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Auger rates for dielectronic recombination cross sections with highly charged relativistic heavy ions
Volume 148, number 8,9
PHYSICS LETTERSA
3 September 1990
Auger rates for dielectronic recombination cross sections with highly charged relativistic heavy ions Peter Z i m m e r e r , N o r b e r t Griin a n d W e r n e r Seheid Instttut fiir Theorettsche Phystk der Justus-Lteblg-Umversltiit, Gtessen, FRG
Received 25 April 1990; rexqsedmanuscript received 28 June 1990;accepted for publication 29 June 1990 Communicatedby B. Fricke
Relativistic wave functions and correctionsto the Coulomb interaction betweentwo electronsare used to calculateAugerrates and the cross section for dielectronic recombinatton (DR). Results are given for hydrogen-likePb ions. The cross section of the DR is compared with an estimate of the cross section of the direct radiative electron capture.
1. Introduction The dielectronic recombination ( D R ) is an important process in high temperature plasma physics [ 1-3 ]. In plasma physics the D R rate coefficients are obtained by integrating the DR cross section over a Maxwellian distribution of the velocity of free electrons in the plasma. Also direct measurements of single D R resonances are possible. At the test storage ring of the MPI in Heidelberg experimental results for the DR cross sections of few-electron atoms were obtained for a limited range of the nuclear charge (Z~< 16) [41. With the comissioning of the new storage ring ESR at GSI in Darmstadt measurements of the DR cross section for high-Z, few-electron ions up to uranium become possible in the near future. An important aspect of these experiments is the study of relativistic effects in connection with the D R process. For example a scaling law has been established for the Zdependence of the D R cross section [ 5,6 ]. One may expect that this sealing law derived for non-relativistic wave functions is inadequate for high Z-values in the relativistic limit. According to our knowledge no relativistic calculations of D R cross sections of hydrogen-like ions with Z > 54 are published in the literature. Nilsen [7] ¢r Supported by BMFT (06 GI 709) and GSI (Darmstadt).
calculated relativistic radiative transition rates and Auger rates for hydrogen-like ions ranging from Ne 9+ to Xe 5a+. Chen and Crasemann evaluated D R rate coefficients for He-like and Be-like ions in the range 6~
2. Cross section for dielectronic recombination The D R cross section from an initial state i to the final state f via an intermediate state d is given in the isolated resonance approximation [ 6 ]: aDs(i--'d-*f) = 2x--'f-2Va(i~d)~0d~d(E,) •
( 1)
Here Pc is the momentum of the incoming electron and E, is the energy of the initial state (we use atomic units h = m = e 2 = 1 ). Va(i~d) is the radiationless capture and excitation probability, OJd is the fluorescence yield of the intermediate state d, and 6d(E,)
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Pubhshers B.V. (North-Holland)
457
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PHYSICS LETTERS A
is a Lorentz curve which determines the shape of the cross section in the vicinity of the resonance: ~d(E,) =
Fd/2g (E,-Ed)2+F2]4"
(2)
Ed is the energy of the intermediate state and Fd its total width, which is given as the sum of the autoionization width F , ( d ) and radiative width F~(d). ogd can be expressed by the partial width A~(d-ff) for the stabilizing decay of the intermediate state d to the final state f:
3 September 1990
The continuum wave function is characterized by the energy ec, parity rtc, and angular momentum quantum number j~ and normalized on the energy scale. The numbers af, nf, and jr denote the quantum numbers of the bound state in the final state, and a,,/-/, J,, and M~ are the quantum numbers of the initial state. For the electron-electron interaction we take a sum of Coulomb and generalized Breit interaction. For two interacting electrons it is given by [ 11 ]
V12 = VIC2"~ Vlg~,
(6)
where A,(d-~f) ogd =
(3)
Fd
Modifications of the fluorescence yield due to cascade effects are not of interest here. The radiationless capture and excitation probability V , ( i o d ) can be expressed by the transition probability of the timereversed Auger process A, (d ~ i ) via the principle of detailed balance, V~(i~d) = ~ A , ( d ~ i ) .
(4)
gd and g, are the statistical weights of the intermediate and initial state, respectively. The factor 2 represents the statistical factor of the free electron. The evaluation of the DR cross section requires the calculation of Auger and radiative rates. For the calculation of relativistic radiative transition rates one can use general purpose programs like the general relativistic atomic structure program (GRASP) of Dyall et al. [ 10 ]. In contrast to the evaluation of the radiative rates there exists no such program for the calculation of relativistic Auger rates.
1 1 VC2- R - ~ l -'rl 2r l
(7)
The Breit interaction is a correction to VC2 due to the exchange of a transverse photon of frequency o9 between the two electrons: V~P~= - , a = , at,a2j
+ OR~,0 ~
o92R
" (8)
Here, ak, is the ith Cartesian component o f the Dirac a-matrices acting on the spinor components of the electron k, and R, is the ith Cartesian component of the distance vector R=r~-r2 between the two electrons. The first term in V t~ is the retarded interaction between two Dirac currents, the second term is the retardation correction to the charge-charge interaction. The initial and final states are expanded in configuration state functions (CSF), ~ ( a f ? t f j f , ecltcjc,JfMf)-'-(~( lSl/2, eclcc; JfMf ) ,
~v(a,II, J,M,)= ~ c,(aOO(7,11,J,M,) .
(9) (10)
r
3. Relmgfisfic Auger rates
Auger transitions are mainly affected by the electron-electron interaction. The transition rate is calculated from perturbation theory and given by A,(i~f) 1 Z M~f Z ] (I//(afTtfJf,eeTteA ; JfMf) I =2X 2J, + 1 st.,~jc
× Vl¥(c~ll, J~M,))12 . 458
(5)
In eq. (10) the ),, characterize the different configurations r. The CSFs for the initial state and its expansion coefficients are obtained from a multiconfiguration Dirac-Fock (MCDF) calculation by using the program of Grant and co-workers [ 12,13 ]. For the bound part of the final state we use a hydrogenic wave function and determine the continuum orbital by solving the radial Dirac-Fock equation with the nuclear potential screened by the bound electron. The exchange term is neglected.
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PHYSICS LETTERS A
4. Results and dis~-assion In o r d e r to study the influence o f relativistic effects on the Auger rates o f all levels o f the doubly occupied L-shell we have done several calculations. The results for the KLL-Auger rates are s u m m a r i z e d in table 1 a n d in fig. 1. The Auger energies a n d the orbital wave functions are o b t a i n e d from an exTable 1 Auger rates from various initial states of the Pbs°+ system m units o f 1012 S - I . Listed are Auger rates calculated with non-relativistic wave functions (NRW), with relativisticwave functions only (RW), with relativistic wave functions and the Breit interaction (RW+BI, co=0) and with relativistic wave functions and the generalized Breit interaction ( RW + GBI, to ~ 0). li)
[2Sl/22pj/2]o [2Si/22Sl/2]o [2st/7.2pt/211 [2Pl/22pl/2 ]o [2sl/22p3/2 ]2 [2pl/22P3/2] 1
NRW
RW
RW+BI
RW+GBI
14
110 593 224 20 23 2 268 166 95 49
595 864 256 55 55 27 222 155 93 84
601 867 258 56 61 31 235 155 101 80
345
14 0
14 0 366 198 0 16
[2pl/22p3/212 [ 2Sl/22p3/2 ] i
[2p3/22P3/2 ]2 [2Pa/22p3/2]0
P b ~*
e*a
<.
-
Auger
t e n d e d average level ( E A L ) calculation with all the ten CSFs o f the d o u b l y occupied L-shell. T h e energies have the relativistic corrections included. T h e orbitals o f the final state are orthogonalized to the initial state orbitals. The overlap between the 1s1/2 arid 2sl/2 wave functions is small ( < 10-3). The same is true for the overlap between continuum and b o u n d orbitals, which is about one o r d e r o f magnitude smaller. The first calculation represents a nonrelativistic l i m i t , o b t a i n e d by scaling the velocity o f light with a factor o f thousand. The second calculation uses only relativistic wave functions b u t no corrections to the C o u l o m b interaction between the electrons. In the t h i r d calculation the usual Breit interaction is applied, which follows from eq. ( 8 ) b y setting t o = 0 , a n d in the last calculation the full correction ( 8 ) is used. The energy levels clearly split up into three groups showing the d o m i n a n c e o f the fine structure splitting o f the one-electron states over the splitting caused b y the interaction between two electrons. F o r the lower levels relativistic wave functions cause a drastic increase o f the Auger rates by a factor o f 3 to 10. The inclusion o f the relativistic corrections to the C o u l o m b interaction increases the rates further. Particularly the [2sz/22pl/2]0 state is strongly affected. F o r the higher levels the corrections are not i m p o r -
rates
400
Table 2 Auger rates for Pbs°+ calculated with relativisttc wave functions and the generalized Brelt interaction. Column 1: no corrections, same values as m column 4 of table 1 (RW+GBI); column 2: corrections of the nonorthogonality between the I s~/2-orbitaland the freely optimized 2st/2-orbital included; column 3: correction due to an optimal level calculation included; column 4: exchange included. The units a r e l012 s - i.
200
li)
1
2
3
4
[2Sl/22pz/2]o
601 867 258 56 61 31 235 155 101 g0
598 864 256 55 60 30 234 153 I00 gI
602 868 258 55 61 31 235 155 101 g0
600 868 258 56 6l 31 235 155 101 80
1000
7
KLL
3 September 1990
g,
0
NRW
i i ~\\\\\~xx\,
RM+BI, w = 0 RW÷GBI, w # 0
mlmB
800 600
m
n¢
~
1
2
3
e
4
5
6
7
N
8
9
10
Level Fig. 1. Auger rates for Pb.°+ for the ten doubly OCCUpied L-shell states as listed in table 1. The rates are shown by vertical bars. From left to right they represent calculations with non-relatiwstic wave functions (NRW), with relativisticwave functions only (RW), w~th relativistic wave functions and the Breit interaction (RW+BI, w=0), and with relativistic wave functions and the generalized Breit interaction ( RW + GBI, to # 0).
[2Sl/22Sl/2]o [2Sl/22Pl/2] I [2pl/22pl/2 ]0 [ 2Sl/22p3/2] 2 [ 2pl/22p3/2] 1 [ 2 p l / 2 2 p 3 / 2 ]2 [ 2Sl/221)3/2 ] 1
[2pa/22p3/2 ]2 [ 2P3/~2P3/2] o
459
Volume 148, number 8,9
PHYSICS LETTERS A
tant. T h e d i f f e r e n c e b e t w e e n the usual Breit intera c t i o n a n d t h e g e n e r a l i z e d Breit i n t e r a c t i o n is q u i t e small. I f o n e c o m p a r e s t h e results o f t h e [ 2s2/2 ] o w i t h n o n r e l a t i v i s t i c c a l c u l a t i o n s f o r Z = 32 [ 14 ] o n e finds that the nonrelativistic rate calculated h e r e has nearly
3 September 1990
t h e s a m e m a g n i t u d e . T h e large v a l u e o f the relativistic rate i n d i c a t e s t h a t the usual scaling law, w h i c h states t h a t the A u g e r rate is i n d e p e n d e n t o f Z, bec o m e s i n v a l i d for the high n u c l e a r charges c o n s i d e r e d here.
Table 3 Cross section for DR at hydrogen-like Pb xons. Listed are the leading configuratmn of the doubly excsted intermediate state d, its energy Ed, total width Fd, the energy ec of the free electron, the Auser width A, of the state d, the leading configuration of the final state f, the energy Edf of the emitted photon, the muRipolarity MP of the stabilization transitions, the partial radmtive width for the decay of the intermediate state d to the final state f, and the energy-averaged cross section for DR defined by # D R ( i - - ' d o f ) = l / A~ Cc l~JrE, E , -+A.,/2 - A ~ / 2 enR (i--, d--, f ) dE,, where Aec is chosen large in comparison to F d.
Ea
Fa
e,
At
(keV)
(eV)
(keV)
(eV)
[2st/22Pl/2]o
-51.710
20.283
49.628
0.396
[2Sl/22sl/2]o
-51.701
16.362
49.638
0.571
[2sl/z2pt/2h
-51.672
20.035
49.666
0.170
[2p,/22p~/2]o
-51.552
23.987
49.786
0.037
[2st/21st/2]l [2Pl/21Sl/2] i [2Sl/21Sl/2]l [2pl/21Sl/2] I [2Pa/21Sl/2]l [2st/21s~/2]l [2pl/21Sl/2] l [2st/21sl/2]o [2pt/21sx/2]o [2sl/21sl/2]l
0.040
[2Pl/21Sl/2] l [2p3/21Sl/2] 1 [2Sl/21St/2] t
Id)
[2Si/22p3/2] 2
-49.093
15.370
52.245
If)
[2p3/21sl/212 [2p3/21St/2] I
--49.038
35.159
52.300
0.020
[2Pl/21S~/211 [2p~/21sl/2]o [2P3/21st/212
[2pl/22p3/2 ]2
-49.020
35.300
52.318
0.155
[2P3/21St/2] t [2pl/21Sl/2] I [2pl/21St/2] 0 [2P3/21St/2] 2
[2p3/21St/2] t [2Sl/22P3/2] 1
-48.987
15.451
52.351
0.102
[2Sl/21St/2] i
[2P3/22p3/2 ]o
460
--46.405
-46.324
30.694
30.698
54.933
55.014
0.067
0.053
75.159 75.026 75.168 75.035 72.450 75.196 75.064 75.000 74.998 75.316 75.184 72.598 77.775 77.579 75.121 75.058 77.698 77.632 75.176 75.113 77.716 77.651 75.194 75.131 77.881
[2p3/21Sl/2] i [2p3/21St/212
77.685 75.226 75.164 77.809
[2p3/21Si/2 ] I
77.746
[2pl/21Sl/2] t [2p3/21Sl/2] 2 [2pa/21SI/2] l
80.412 77.890 77.827
[2Sl/21Si/2] 0 [2p3/21SI/212
[2p3/z2p3/2 ] 2
MP
Ar (eV)
ffmtAec (kb eV)
E1 M1 M1 E1 El El M1 E1 M1 M1 El E1 E1 M2 M2 MI MI E1 M2 El E1 E1 El M2 M2 E1 El El M2 E1 M1 MI El M2 El M2
19.854 0.034 0.041 15.744 0.006 13.646 0.023 6.185 0.011 0.027 23.915 0.008 15.227 0.041 0.027 0.017 0.017 5.125 0.068 10.137 16.541 3.252 15.061 0.041 0.027 10.157 9.844 4.660 0.069 10.565 0.029 0.006 14.995 0.097 15.464 0.041 0 002 0.138 30.477
13.999 0.024 0.052 19.837 0.007 12.511 0.021 5.671 0.010 0.002 1.332 0.000 6.801 0.018 0.012 0.008 0.008 0.302 0.004 0.597 0.974 0.191 11.288 0.031 0.021 7.612 7.378 3.145 0.047 7.132 0.019 0.004 5.282 0.034 5.447 0.014 0.000 0.008 1.695
(keV)
[2S1/21Sl/2]o
[2Pj/z2p3/2] j
Ear
El
M2 El
Volume 148, number 8,9
PHYSICSLETTERSA
For resonant transfer excitation (RTE) experiments the [2Sl/22pl/2]t state is important, which can decay via an electric dipole transition into the [1St/22S1/2]o s t a t e . The latter state decays exclusively by emission of two photons. With the detection of these two photons one can measure the cross section of a single D R resonance in a RTE experiment [5 ]. In order to examine the influence of the nonorthogonality between the bound orbitals we have included all relevant configurations of the type [ lst/22K] in the expansion of the initial state. The 1st n-wave function is of hydrogenic type and the 2K wave functions are obtained from a MCDF calculation with fixed 1Sl/2 w a v e function. This procedure leads to Auger rates which differ from that of table I by less than 2% (see column 2 in table 2). The corrections due to an optimal level (OL) calculation are of the same magnitude (column 3 in table 2). Also the influence of the exchange interaction on the continuum orbitals was checked. To obtain continuum orbitals with the exchange interaction included we solved the Dirac-Fock equations twice. In the first step continuum orbitals were calculated with the exchange potentials neglected. With these continuum orbitals we determined the exchange potential for the second step and solved the Dirac-Fock equations again. The last step affects the Auger rates at most by 1% as shown in column 4 of table 2. The radiative decay rates of the intermediate states d, which are needed for the D R cross section ( 1 ) are obtained from the GRASP program. The relevant transition energies and rates for the DR are summarized in table 3. The total cross section for the capture of the free electron via D R is shown in fig. 2. In fig. 3 we compare the D R cross section with an estimated cross section for the direct radiative electron capture (REC) calculated with hydrogenic wave functions for the bound and continuum electron at a nuclear charge of Z = 82. For the REC curve only dipole transitions were considered. The figure shows that in the case of heavier ions both cross sections can be of the same order of magnitude. Therefore, the usual treatment of the REC as a uniform background signal small compared to the D R resonances and of neglecting the interference term in the cross
3 September 1990 I
I
I
e" + P b "
1.2
I
i
I
- > (Pbm*)" - > (Pbm*)" + 7
1.0 ,.Q
0.8-
V
0.6b
0.4
0.2 . ~ 0.0
I
I
49.6
,
52.2
49.8
I
I
I
52.4 54.8
55.0
e~ ( k e V ) Fig. 2. Total cross section for capture of a free electron via DR by Pb s~* as a functmn of the energy of the electron. The three curves are calculated with different Auger rates. The full curve is o13tained from a calculation with relativistic wave functions and the Breit interacUon, the dashed curve with relativistic wave functmns only, and the dotted curve with non-relativistic wave functions. The resonance energies are taken to be the same in these calculations.
2
1
I
I
e- + P b ' * - >
1
I
I
(Pb'e*) "" - >
I
DR REC
0.5 v
I
(Pb'e*)" + T
0.2 o.1 0.05
0.02 i
49
I
I
I
f
I
50
51
52
53
54
55
eo ( k e V ) Ftg. 3. Comparison of the cross section for capture of a free electron via DR (full curve) or REC (dotted curve) as function of the energy of the electron. The electron zs captured to a Pb sl + Ion.
section may not be correct [ 6 ]. A more precise answer to that question can be obtained by a unified treatment of the D R and the REC [ 15,16 ].
5. S u m m a r y
We have shown that the relativistic corrections to the Auger rates lead to a drastic increase of the D R 461
Volume 148, number 8,9
PHYSICS LETTERS A
cross section for some KLL-resonances o f the hydrogen-like Pb ion. In that context the effects o f relativistic wave functions a n d o f the Breit corrections to the C o u l o m b interaction between the electrons were examined. The calculated rates are partly 2 to 3 times higher as one would expect from a non-relativistic scaling law [6 ]. A c o m p a r i s o n o f the D R cross section with that o f the R E C process shows that interference p h e n o m e n a between the two processes b e c o m e important. W o r k in this direction is now in progress. Very recently Chen published a calculation o f RTE cross sections for U sg+ ions, which shows a similar increase due to relativistic effects on the Auger rates [17].
References [1 ] M.J. Seaton and P.J. Storey, in: Atomic processes and applications, eds. P.G. Burke and B.L. Moiselwitsch (NorthHolland, Amsterdam, 1976)p. 133. [ 2 ] A. Burgress, Astrophys. J. 139 (1964) 776
462
3 September 1990
[ 3 ] A. Burgress, Astrophys. J. 141 (1965) 1589. [4] G. Kilgus, J. Berger, P. Blatl, M. Grieser, D. Hahs, B. Hochadel, E. Jaeschke, D. Kramer, R. Neumann, G. Neureither, W. Ott, D. Schwalm, M. Steck, R. Stokstad, E. Szmola, A. Wolf, R. Schuch, A. Mtiller and M. Wagner, Phys. Rev. Lett. 64 (1990) 737. [ 5 ] S. Reusch, doctoral thesis, University of Glessen (1988). [6] Y. Hahn, Adv. At. MoL Phys. 21 (1985) 123. [ 7 ] J. Nilsen, At. Data Nucl. Data Tables 37 (1987) 191. [8] M.H. Chen, Phys. Rev. A 33 (1986) 994. [9] M.H. Cben and B. Crasemann, Phys. Rev. A 38 (1988) 5595. [ 10] ICG. Dyall, I.P. Grant, C.T. Johnson, E.P. Plummer and F. Parpia, Comput. Phys. Commun. 55 (1988) 425. [ 11 ] J.B. Mann and W.R. Johnson, Phys. Rev. A 4 ( 1971 ) 41. [ 12 ] I.P. Grant, B.J. McKenzie, P.H. Norrington, D.F. Mayers and N.C. Pyper, Comput. Phys. Commun. 21 (1980) 207. [ 13] B.J. McKenzie, I.P. Grant and P.H. Norrington, Comput. Phys. Commun. 21 (1980) 233. [ 14] L.A. Vamshtem and U.1. Safronova, At. Data Nucl. Data Tables 21 (1978) 49. [ 15 ] G. Alber, J. Cooper and A.R.P. Rau, Phys. Rev. A 30 (1984) 2845 [16]V.L. Jacobs, J. Cooper and S.L Haan, Phys. Rev. A 36 (1987) 1093. [ 17 ] M.H. Chen, Phys. Rev. A 41 (1990) 4102.
PHYSICS LETTERSA
3 September 1990
Auger rates for dielectronic recombination cross sections with highly charged relativistic heavy ions Peter Z i m m e r e r , N o r b e r t Griin a n d W e r n e r Seheid Instttut fiir Theorettsche Phystk der Justus-Lteblg-Umversltiit, Gtessen, FRG
Received 25 April 1990; rexqsedmanuscript received 28 June 1990;accepted for publication 29 June 1990 Communicatedby B. Fricke
Relativistic wave functions and correctionsto the Coulomb interaction betweentwo electronsare used to calculateAugerrates and the cross section for dielectronic recombinatton (DR). Results are given for hydrogen-likePb ions. The cross section of the DR is compared with an estimate of the cross section of the direct radiative electron capture.
1. Introduction The dielectronic recombination ( D R ) is an important process in high temperature plasma physics [ 1-3 ]. In plasma physics the D R rate coefficients are obtained by integrating the DR cross section over a Maxwellian distribution of the velocity of free electrons in the plasma. Also direct measurements of single D R resonances are possible. At the test storage ring of the MPI in Heidelberg experimental results for the DR cross sections of few-electron atoms were obtained for a limited range of the nuclear charge (Z~< 16) [41. With the comissioning of the new storage ring ESR at GSI in Darmstadt measurements of the DR cross section for high-Z, few-electron ions up to uranium become possible in the near future. An important aspect of these experiments is the study of relativistic effects in connection with the D R process. For example a scaling law has been established for the Zdependence of the D R cross section [ 5,6 ]. One may expect that this sealing law derived for non-relativistic wave functions is inadequate for high Z-values in the relativistic limit. According to our knowledge no relativistic calculations of D R cross sections of hydrogen-like ions with Z > 54 are published in the literature. Nilsen [7] ¢r Supported by BMFT (06 GI 709) and GSI (Darmstadt).
calculated relativistic radiative transition rates and Auger rates for hydrogen-like ions ranging from Ne 9+ to Xe 5a+. Chen and Crasemann evaluated D R rate coefficients for He-like and Be-like ions in the range 6~
2. Cross section for dielectronic recombination The D R cross section from an initial state i to the final state f via an intermediate state d is given in the isolated resonance approximation [ 6 ]: aDs(i--'d-*f) = 2x--'f-2Va(i~d)~0d~d(E,) •
( 1)
Here Pc is the momentum of the incoming electron and E, is the energy of the initial state (we use atomic units h = m = e 2 = 1 ). Va(i~d) is the radiationless capture and excitation probability, OJd is the fluorescence yield of the intermediate state d, and 6d(E,)
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Pubhshers B.V. (North-Holland)
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PHYSICS LETTERS A
is a Lorentz curve which determines the shape of the cross section in the vicinity of the resonance: ~d(E,) =
Fd/2g (E,-Ed)2+F2]4"
(2)
Ed is the energy of the intermediate state and Fd its total width, which is given as the sum of the autoionization width F , ( d ) and radiative width F~(d). ogd can be expressed by the partial width A~(d-ff) for the stabilizing decay of the intermediate state d to the final state f:
3 September 1990
The continuum wave function is characterized by the energy ec, parity rtc, and angular momentum quantum number j~ and normalized on the energy scale. The numbers af, nf, and jr denote the quantum numbers of the bound state in the final state, and a,,/-/, J,, and M~ are the quantum numbers of the initial state. For the electron-electron interaction we take a sum of Coulomb and generalized Breit interaction. For two interacting electrons it is given by [ 11 ]
V12 = VIC2"~ Vlg~,
(6)
where A,(d-~f) ogd =
(3)
Fd
Modifications of the fluorescence yield due to cascade effects are not of interest here. The radiationless capture and excitation probability V , ( i o d ) can be expressed by the transition probability of the timereversed Auger process A, (d ~ i ) via the principle of detailed balance, V~(i~d) = ~ A , ( d ~ i ) .
(4)
gd and g, are the statistical weights of the intermediate and initial state, respectively. The factor 2 represents the statistical factor of the free electron. The evaluation of the DR cross section requires the calculation of Auger and radiative rates. For the calculation of relativistic radiative transition rates one can use general purpose programs like the general relativistic atomic structure program (GRASP) of Dyall et al. [ 10 ]. In contrast to the evaluation of the radiative rates there exists no such program for the calculation of relativistic Auger rates.
1 1 VC2- R - ~ l -'rl 2r l
(7)
The Breit interaction is a correction to VC2 due to the exchange of a transverse photon of frequency o9 between the two electrons: V~P~= - , a = , at,a2j
+ OR~,0 ~
o92R
" (8)
Here, ak, is the ith Cartesian component o f the Dirac a-matrices acting on the spinor components of the electron k, and R, is the ith Cartesian component of the distance vector R=r~-r2 between the two electrons. The first term in V t~ is the retarded interaction between two Dirac currents, the second term is the retardation correction to the charge-charge interaction. The initial and final states are expanded in configuration state functions (CSF), ~ ( a f ? t f j f , ecltcjc,JfMf)-'-(~( lSl/2, eclcc; JfMf ) ,
~v(a,II, J,M,)= ~ c,(aOO(7,11,J,M,) .
(9) (10)
r
3. Relmgfisfic Auger rates
Auger transitions are mainly affected by the electron-electron interaction. The transition rate is calculated from perturbation theory and given by A,(i~f) 1 Z M~f Z ] (I//(afTtfJf,eeTteA ; JfMf) I =2X 2J, + 1 st.,~jc
× Vl¥(c~ll, J~M,))12 . 458
(5)
In eq. (10) the ),, characterize the different configurations r. The CSFs for the initial state and its expansion coefficients are obtained from a multiconfiguration Dirac-Fock (MCDF) calculation by using the program of Grant and co-workers [ 12,13 ]. For the bound part of the final state we use a hydrogenic wave function and determine the continuum orbital by solving the radial Dirac-Fock equation with the nuclear potential screened by the bound electron. The exchange term is neglected.
Volume 148, number 8,9
PHYSICS LETTERS A
4. Results and dis~-assion In o r d e r to study the influence o f relativistic effects on the Auger rates o f all levels o f the doubly occupied L-shell we have done several calculations. The results for the KLL-Auger rates are s u m m a r i z e d in table 1 a n d in fig. 1. The Auger energies a n d the orbital wave functions are o b t a i n e d from an exTable 1 Auger rates from various initial states of the Pbs°+ system m units o f 1012 S - I . Listed are Auger rates calculated with non-relativistic wave functions (NRW), with relativisticwave functions only (RW), with relativistic wave functions and the Breit interaction (RW+BI, co=0) and with relativistic wave functions and the generalized Breit interaction ( RW + GBI, to ~ 0). li)
[2Sl/22pj/2]o [2Si/22Sl/2]o [2st/7.2pt/211 [2Pl/22pl/2 ]o [2sl/22p3/2 ]2 [2pl/22P3/2] 1
NRW
RW
RW+BI
RW+GBI
14
110 593 224 20 23 2 268 166 95 49
595 864 256 55 55 27 222 155 93 84
601 867 258 56 61 31 235 155 101 80
345
14 0
14 0 366 198 0 16
[2pl/22p3/212 [ 2Sl/22p3/2 ] i
[2p3/22P3/2 ]2 [2Pa/22p3/2]0
P b ~*
e*a
<.
-
Auger
t e n d e d average level ( E A L ) calculation with all the ten CSFs o f the d o u b l y occupied L-shell. T h e energies have the relativistic corrections included. T h e orbitals o f the final state are orthogonalized to the initial state orbitals. The overlap between the 1s1/2 arid 2sl/2 wave functions is small ( < 10-3). The same is true for the overlap between continuum and b o u n d orbitals, which is about one o r d e r o f magnitude smaller. The first calculation represents a nonrelativistic l i m i t , o b t a i n e d by scaling the velocity o f light with a factor o f thousand. The second calculation uses only relativistic wave functions b u t no corrections to the C o u l o m b interaction between the electrons. In the t h i r d calculation the usual Breit interaction is applied, which follows from eq. ( 8 ) b y setting t o = 0 , a n d in the last calculation the full correction ( 8 ) is used. The energy levels clearly split up into three groups showing the d o m i n a n c e o f the fine structure splitting o f the one-electron states over the splitting caused b y the interaction between two electrons. F o r the lower levels relativistic wave functions cause a drastic increase o f the Auger rates by a factor o f 3 to 10. The inclusion o f the relativistic corrections to the C o u l o m b interaction increases the rates further. Particularly the [2sz/22pl/2]0 state is strongly affected. F o r the higher levels the corrections are not i m p o r -
rates
400
Table 2 Auger rates for Pbs°+ calculated with relativisttc wave functions and the generalized Brelt interaction. Column 1: no corrections, same values as m column 4 of table 1 (RW+GBI); column 2: corrections of the nonorthogonality between the I s~/2-orbitaland the freely optimized 2st/2-orbital included; column 3: correction due to an optimal level calculation included; column 4: exchange included. The units a r e l012 s - i.
200
li)
1
2
3
4
[2Sl/22pz/2]o
601 867 258 56 61 31 235 155 101 g0
598 864 256 55 60 30 234 153 I00 gI
602 868 258 55 61 31 235 155 101 g0
600 868 258 56 6l 31 235 155 101 80
1000
7
KLL
3 September 1990
g,
0
NRW
i i ~\\\\\~xx\,
RM+BI, w = 0 RW÷GBI, w # 0
mlmB
800 600
m
n¢
~
1
2
3
e
4
5
6
7
N
8
9
10
Level Fig. 1. Auger rates for Pb.°+ for the ten doubly OCCUpied L-shell states as listed in table 1. The rates are shown by vertical bars. From left to right they represent calculations with non-relatiwstic wave functions (NRW), with relativisticwave functions only (RW), w~th relativistic wave functions and the Breit interaction (RW+BI, w=0), and with relativistic wave functions and the generalized Breit interaction ( RW + GBI, to # 0).
[2Sl/22Sl/2]o [2Sl/22Pl/2] I [2pl/22pl/2 ]0 [ 2Sl/22p3/2] 2 [ 2pl/22p3/2] 1 [ 2 p l / 2 2 p 3 / 2 ]2 [ 2Sl/221)3/2 ] 1
[2pa/22p3/2 ]2 [ 2P3/~2P3/2] o
459
Volume 148, number 8,9
PHYSICS LETTERS A
tant. T h e d i f f e r e n c e b e t w e e n the usual Breit intera c t i o n a n d t h e g e n e r a l i z e d Breit i n t e r a c t i o n is q u i t e small. I f o n e c o m p a r e s t h e results o f t h e [ 2s2/2 ] o w i t h n o n r e l a t i v i s t i c c a l c u l a t i o n s f o r Z = 32 [ 14 ] o n e finds that the nonrelativistic rate calculated h e r e has nearly
3 September 1990
t h e s a m e m a g n i t u d e . T h e large v a l u e o f the relativistic rate i n d i c a t e s t h a t the usual scaling law, w h i c h states t h a t the A u g e r rate is i n d e p e n d e n t o f Z, bec o m e s i n v a l i d for the high n u c l e a r charges c o n s i d e r e d here.
Table 3 Cross section for DR at hydrogen-like Pb xons. Listed are the leading configuratmn of the doubly excsted intermediate state d, its energy Ed, total width Fd, the energy ec of the free electron, the Auser width A, of the state d, the leading configuration of the final state f, the energy Edf of the emitted photon, the muRipolarity MP of the stabilization transitions, the partial radmtive width for the decay of the intermediate state d to the final state f, and the energy-averaged cross section for DR defined by # D R ( i - - ' d o f ) = l / A~ Cc l~JrE, E , -+A.,/2 - A ~ / 2 enR (i--, d--, f ) dE,, where Aec is chosen large in comparison to F d.
Ea
Fa
e,
At
(keV)
(eV)
(keV)
(eV)
[2st/22Pl/2]o
-51.710
20.283
49.628
0.396
[2Sl/22sl/2]o
-51.701
16.362
49.638
0.571
[2sl/z2pt/2h
-51.672
20.035
49.666
0.170
[2p,/22p~/2]o
-51.552
23.987
49.786
0.037
[2st/21st/2]l [2Pl/21Sl/2] i [2Sl/21Sl/2]l [2pl/21Sl/2] I [2Pa/21Sl/2]l [2st/21s~/2]l [2pl/21Sl/2] l [2st/21sl/2]o [2pt/21sx/2]o [2sl/21sl/2]l
0.040
[2Pl/21Sl/2] l [2p3/21Sl/2] 1 [2Sl/21St/2] t
Id)
[2Si/22p3/2] 2
-49.093
15.370
52.245
If)
[2p3/21sl/212 [2p3/21St/2] I
--49.038
35.159
52.300
0.020
[2Pl/21S~/211 [2p~/21sl/2]o [2P3/21st/212
[2pl/22p3/2 ]2
-49.020
35.300
52.318
0.155
[2P3/21St/2] t [2pl/21Sl/2] I [2pl/21St/2] 0 [2P3/21St/2] 2
[2p3/21St/2] t [2Sl/22P3/2] 1
-48.987
15.451
52.351
0.102
[2Sl/21St/2] i
[2P3/22p3/2 ]o
460
--46.405
-46.324
30.694
30.698
54.933
55.014
0.067
0.053
75.159 75.026 75.168 75.035 72.450 75.196 75.064 75.000 74.998 75.316 75.184 72.598 77.775 77.579 75.121 75.058 77.698 77.632 75.176 75.113 77.716 77.651 75.194 75.131 77.881
[2p3/21Sl/2] i [2p3/21St/212
77.685 75.226 75.164 77.809
[2p3/21Si/2 ] I
77.746
[2pl/21Sl/2] t [2p3/21Sl/2] 2 [2pa/21SI/2] l
80.412 77.890 77.827
[2Sl/21Si/2] 0 [2p3/21SI/212
[2p3/z2p3/2 ] 2
MP
Ar (eV)
ffmtAec (kb eV)
E1 M1 M1 E1 El El M1 E1 M1 M1 El E1 E1 M2 M2 MI MI E1 M2 El E1 E1 El M2 M2 E1 El El M2 E1 M1 MI El M2 El M2
19.854 0.034 0.041 15.744 0.006 13.646 0.023 6.185 0.011 0.027 23.915 0.008 15.227 0.041 0.027 0.017 0.017 5.125 0.068 10.137 16.541 3.252 15.061 0.041 0.027 10.157 9.844 4.660 0.069 10.565 0.029 0.006 14.995 0.097 15.464 0.041 0 002 0.138 30.477
13.999 0.024 0.052 19.837 0.007 12.511 0.021 5.671 0.010 0.002 1.332 0.000 6.801 0.018 0.012 0.008 0.008 0.302 0.004 0.597 0.974 0.191 11.288 0.031 0.021 7.612 7.378 3.145 0.047 7.132 0.019 0.004 5.282 0.034 5.447 0.014 0.000 0.008 1.695
(keV)
[2S1/21Sl/2]o
[2Pj/z2p3/2] j
Ear
El
M2 El
Volume 148, number 8,9
PHYSICSLETTERSA
For resonant transfer excitation (RTE) experiments the [2Sl/22pl/2]t state is important, which can decay via an electric dipole transition into the [1St/22S1/2]o s t a t e . The latter state decays exclusively by emission of two photons. With the detection of these two photons one can measure the cross section of a single D R resonance in a RTE experiment [5 ]. In order to examine the influence of the nonorthogonality between the bound orbitals we have included all relevant configurations of the type [ lst/22K] in the expansion of the initial state. The 1st n-wave function is of hydrogenic type and the 2K wave functions are obtained from a MCDF calculation with fixed 1Sl/2 w a v e function. This procedure leads to Auger rates which differ from that of table I by less than 2% (see column 2 in table 2). The corrections due to an optimal level (OL) calculation are of the same magnitude (column 3 in table 2). Also the influence of the exchange interaction on the continuum orbitals was checked. To obtain continuum orbitals with the exchange interaction included we solved the Dirac-Fock equations twice. In the first step continuum orbitals were calculated with the exchange potentials neglected. With these continuum orbitals we determined the exchange potential for the second step and solved the Dirac-Fock equations again. The last step affects the Auger rates at most by 1% as shown in column 4 of table 2. The radiative decay rates of the intermediate states d, which are needed for the D R cross section ( 1 ) are obtained from the GRASP program. The relevant transition energies and rates for the DR are summarized in table 3. The total cross section for the capture of the free electron via D R is shown in fig. 2. In fig. 3 we compare the D R cross section with an estimated cross section for the direct radiative electron capture (REC) calculated with hydrogenic wave functions for the bound and continuum electron at a nuclear charge of Z = 82. For the REC curve only dipole transitions were considered. The figure shows that in the case of heavier ions both cross sections can be of the same order of magnitude. Therefore, the usual treatment of the REC as a uniform background signal small compared to the D R resonances and of neglecting the interference term in the cross
3 September 1990 I
I
I
e" + P b "
1.2
I
i
I
- > (Pbm*)" - > (Pbm*)" + 7
1.0 ,.Q
0.8-
V
0.6b
0.4
0.2 . ~ 0.0
I
I
49.6
,
52.2
49.8
I
I
I
52.4 54.8
55.0
e~ ( k e V ) Fig. 2. Total cross section for capture of a free electron via DR by Pb s~* as a functmn of the energy of the electron. The three curves are calculated with different Auger rates. The full curve is o13tained from a calculation with relativistic wave functions and the Breit interacUon, the dashed curve with relativistic wave functmns only, and the dotted curve with non-relativistic wave functions. The resonance energies are taken to be the same in these calculations.
2
1
I
I
e- + P b ' * - >
1
I
I
(Pb'e*) "" - >
I
DR REC
0.5 v
I
(Pb'e*)" + T
0.2 o.1 0.05
0.02 i
49
I
I
I
f
I
50
51
52
53
54
55
eo ( k e V ) Ftg. 3. Comparison of the cross section for capture of a free electron via DR (full curve) or REC (dotted curve) as function of the energy of the electron. The electron zs captured to a Pb sl + Ion.
section may not be correct [ 6 ]. A more precise answer to that question can be obtained by a unified treatment of the D R and the REC [ 15,16 ].
5. S u m m a r y
We have shown that the relativistic corrections to the Auger rates lead to a drastic increase of the D R 461
Volume 148, number 8,9
PHYSICS LETTERS A
cross section for some KLL-resonances o f the hydrogen-like Pb ion. In that context the effects o f relativistic wave functions a n d o f the Breit corrections to the C o u l o m b interaction between the electrons were examined. The calculated rates are partly 2 to 3 times higher as one would expect from a non-relativistic scaling law [6 ]. A c o m p a r i s o n o f the D R cross section with that o f the R E C process shows that interference p h e n o m e n a between the two processes b e c o m e important. W o r k in this direction is now in progress. Very recently Chen published a calculation o f RTE cross sections for U sg+ ions, which shows a similar increase due to relativistic effects on the Auger rates [17].
References [1 ] M.J. Seaton and P.J. Storey, in: Atomic processes and applications, eds. P.G. Burke and B.L. Moiselwitsch (NorthHolland, Amsterdam, 1976)p. 133. [ 2 ] A. Burgress, Astrophys. J. 139 (1964) 776
462
3 September 1990
[ 3 ] A. Burgress, Astrophys. J. 141 (1965) 1589. [4] G. Kilgus, J. Berger, P. Blatl, M. Grieser, D. Hahs, B. Hochadel, E. Jaeschke, D. Kramer, R. Neumann, G. Neureither, W. Ott, D. Schwalm, M. Steck, R. Stokstad, E. Szmola, A. Wolf, R. Schuch, A. Mtiller and M. Wagner, Phys. Rev. Lett. 64 (1990) 737. [ 5 ] S. Reusch, doctoral thesis, University of Glessen (1988). [6] Y. Hahn, Adv. At. MoL Phys. 21 (1985) 123. [ 7 ] J. Nilsen, At. Data Nucl. Data Tables 37 (1987) 191. [8] M.H. Chen, Phys. Rev. A 33 (1986) 994. [9] M.H. Cben and B. Crasemann, Phys. Rev. A 38 (1988) 5595. [ 10] ICG. Dyall, I.P. Grant, C.T. Johnson, E.P. Plummer and F. Parpia, Comput. Phys. Commun. 55 (1988) 425. [ 11 ] J.B. Mann and W.R. Johnson, Phys. Rev. A 4 ( 1971 ) 41. [ 12 ] I.P. Grant, B.J. McKenzie, P.H. Norrington, D.F. Mayers and N.C. Pyper, Comput. Phys. Commun. 21 (1980) 207. [ 13] B.J. McKenzie, I.P. Grant and P.H. Norrington, Comput. Phys. Commun. 21 (1980) 233. [ 14] L.A. Vamshtem and U.1. Safronova, At. Data Nucl. Data Tables 21 (1978) 49. [ 15 ] G. Alber, J. Cooper and A.R.P. Rau, Phys. Rev. A 30 (1984) 2845 [16]V.L. Jacobs, J. Cooper and S.L Haan, Phys. Rev. A 36 (1987) 1093. [ 17 ] M.H. Chen, Phys. Rev. A 41 (1990) 4102.