- Email: [email protected]
Evaluation of configuration-interaction effects on atomic transition arrays
J. Quanc.
Pergamon
Specwosc. Radiar.
0022-4073(95)00039-9
Transfer Vol. 54, No. l/2. pp. 43-51, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-4073195 $9.50 + 0.00
EVALUATION OF CONFIGURATION-INTERACTION EFFECTS ON ATOMIC TRANSITION ARRAYS J. BAUCHE,_F C. BAUCHE-ARNOULT,?
A. BACHELIER,?
and
W. H. GOLDSTEIN1 tLaboratoire
Aimt
Cotton,
CNRS II, 91405 Orsay, France and iLawrence Laboratory, Livermore, CA 94551, U.S.A.
Livermore
National
Abstract-In the field of the ab-initio simulations of atomic spectra, a method is described for taking into account the configuration-interaction effects on electric-dipolar transition arrays. As an application, the strong effects observed in the 4f-4d’ arrays of the spectra of Praseodymium XV through XVIII are interpreted quantitatively. An original method is proposed for computing the intensities of the “forbidden” electric-dipolar arrays, which are generated by configuration interaction.
1. INTRODUCTION
We report here current results obtained in the evaluation of global effects of configuration interaction in atomic spectra. Global means that the effects studied relate to groups of levels (the electronic configurations) and of radiative lines (the transition arrays), but not to individual levels or lines.’ If configuration interaction is considered as a small perturbation of the central-field scheme, its effects on energy levels and line intensities are, at most, of the second and first orders, respectively. Dramatic first-order changes in the intensities of some transition arrays have already been observed, and interpreted. 2,3In the present work, that method of interpretation is generalized to more complex situations, and results from a computer code are presented. Another method is proposed for the case where the intensities are only perturbed to the second order. In the following, the words interaction and mixing are equivalent, because the perturbations studied are actually generated by the mixing of the quantum states of different configurations. The configurations considered are those defined by Condon and Shortley.4 They are not the relativistic (sub-) configurations whose mixing has been treated elsewhere.5
2. WHAT
IS KNOWN
UP TO NOW
Let us consider three electronic configurations C,, C,, C,, linked by the electric&polar transition arrays C,--C, and C,-C,, denoted C,-(C, + C,) altogether (Fig. 1). If the levels of configurations C, and C, are mixed through the electrostatic interelectronic-repulsion operator G, the intensities of the radiative lines are changed. In the following, we are essentially interested in the quantum part of the line intensities, i.e., in the line strengths, and in the total array strengths, which are the sums of the corresponding line strengths.6 The strength of a line is the square of its amplitude, and can be written, in tensor-operator form
for the line between the levels a,J, and azJ2. The LXsymbols refer to the intermediate couplings in the relevant configurations. D(I) is the electric-dipolar tensor operator. For the effects of configuration mixing on the array strengths, there appear two different situations. 43
J. Bauche et al
44
2.1. First -order array mixing
What we call the first-order mixing can be characterized by two properties, which are equivalent. (i) The energy W of the center of gravity of the energies of all the lines of both arrays, weighted by their strengths, is changed by the mixing. (ii) If the C, and C, configurations lie far apart one from the other, the changes of the line strengths, and of W, are linear functions of the Rk Slater integrals responsible for the mixing. Example: the dN + ,s_(dN + ’p + dNsp) mixing, in the transition-metal spectra. 2.2. Second-order array mixing In the case of a second-order array mixing, the properties (i)+ii) above are replaced by two other ones. (i) The center of gravity W is not changed by the mixing. (ii) If the C, and C, configurations lie far apart one from the other, the changes of the line strengths, and of W, are quadratic functions of the Rk Slater integrals responsible for the mixing. These properties are also valid for the cases where the electric-dipolar transition array between C3 and one of the other configurations is said to be “forbidden”, i.e. when it is generated by the mixing. Example: the dNs2-(dN+ ‘p + dNsp) mixing, in the transition-metal spectra. 2.3. Computation of the$rst-order
changes in the total strengths of the arrays
A matrix method has been proposed a few years ago for the computation of the first-order changes in the array strengths.’ The matrix in Fig. 1, corresponding to the C3-(C, + C,) case, is called a triad. In the energy matrix, the diagonal elements E, , E, and E, are the weighted average energies of the levels of the configurations C, , C, and C,, respectively, in the absence of mixing. The weight of each level is its emissivity, defined as the sum of the strengths of all the lines related to this level. E,, E2 and E3 are also called the average energies of the emissive zones.6 The off-diagonal elements a,, and a23 are the total amplitudes of the arrays C,--C, and C,-C,, respectively, defined as the positive square roots of their total strengths. All these quantities can be computed using compact formulas.’ The off-diagonal energy matrix element El2 is defined in the following way. When the energy matrix is diagonalized, with eigenvalues denoted E; and E;, its eigenvectors (a, /3) and (j?, -a) are used for obtaining the transformed strengths s;~ = (u;~)~ and s& = (a;,)2, with ai = UZ,~+ flu23and a& = Ba,3 - ua23. Thus, in the occurrence of the mixing, the value of the weighted center of gravity of the array energies reads W’=
[s;jE; +s;,E;]/[s;J
+s;J
- E3.
(2)
The general formal expression of E,, has been derived in order that W’ be equal to the weighted center of gravity of the line energies, as obtained through a detailed line-by-line calculation.’ Using this expression also ensures that the transformed strengths are exact to the first order with respect to the Rk Slater integrals, i.e. when these integrals are infinitely small.
3. A MORE
GENERAL
METHOD
FOR
COMPUTING
THE
FIRST-ORDER
MIXING
The matrix in Fig. 2 corresponds to the case denoted (C, + C,) - (C, + C,), called a tetrad, where two even configurations, C, and C,, are both linked by electricdipolar transitions with two odd ones, C, and C,. This case is more complicated than that of a triad, but it is not rare in the spectra. The energy and amplitude matrix elements are computed as it is recalled in Sec. 2.3.
of configuration-interaction
effects
I
I E,
Cl
1
Et2
1
Q13
Fig. 1. Matrix for computing the case of the triad C,
After all four energy matrices are diagonalized, the transformed strengths si3 through siq are exact to the first order with respect to the Rk integrals. However, in general, their sum is not rigorously equal to &, =
s13 +
s23 +
sl4
+
(3)
s24
the value before the diagonalization, in contrast with the detailed line-by-line method. This is due to the fact that, in the matrix method used, the occurrence of higher-order contributions is not taken into account. Furthermore, all the elements of the 4 x 4 off-diagonal amplitude matrix in Fig. 2 are now different from zero.
c2
Fig. 2. Matrix for computing the case of the tetrad (C, + C&o, + Cz). A tetrad is the combination of 4 intermingled triads. This is the most general case, where each configuration has one emissive zone relating to each configuration of the other parity. For example, E, (resp. H,) is the energy of the C, emissive zone relating to the emission towards C3 (resp. C,). In this figure, the energy order of the 8 emissive zones is arbitrary.
J. Bauche et al
46
Two approximate
methods are proposed for correcting this drawback. (i) In the first method, one simply renormalizes the strengths in order that their sum be equal to &0,. (ii) The second method uses the eigenvectors in a different way. In the first step, one transforms the initial amplitudes by means of the eigenvectors of the upper two (odd) matrices in Fig. 2; then, one transforms the results by means of the eigenvectors of the even matrices. This yields a set of four strengths whose sum is equal to S,,,. The second step is identical to the first one, but for the order in which the odd and even eigenvectors are used. The final values of the strengths are the half-sums of the results of the two steps. It can be proved that any spectrum is the association of many intermingled triads and tetrads (no more complicated case can occur), if one discards the mixing between configurations differing only by the principal quantum number n of one orbital. The latter simplification is valid for the HartreeFock radial functions, which are known to be fairly close to the central-field parametricpotential functions. In each case, even and odd matrices are built like those in Fig. 2. Each transition array relates to one even and one odd basis vector. Therefore, the orders of the whole even and odd matrices are both equal to the number of arrays. 4. THE
CODE
FOR
COMPUTING
FIRST-ORDER
MIXING
EFFECTS
A computer code has been written for computing the first-order mixing effects on the array strengths of a given spectrum. The input data file contains the list of the configurations to be considered. All the corresponding electricdipolar transition arrays are automatically taken into account in the calculation. The first step is the use of the ab-initio relativistic parametric-potential code RELAC8 which generates all the needed radial integrals, and deduces the average energies of the configurations, and of their emissive zones. In the second step, the even and odd configurations are implemented, and diagonalized. No large-scale diagonalization is required, because both matrices break into many block-diagonal smaller matrices. In Fig. 2, both 4 x 4 energy matrices break into two 2 x 2 submatrices. In the Pr calculations (Section 5), the largest submatrix which we have encountered was 7 x 7. Then, the eigenvectors are used for computing the transformed strengths, in the two ways described in Section 3. The size limitations of the code are essentially those of RELAC, for the maximum number of configurations which can be treated. Otherwise, the number of arrays is not limited. 5. EXAMPLE:
THE
IONS
PrXV
THROUGH
PrXVIII
A part of the experimental praseodymium spectrum generated by a high-voltage vacuum spark, published by Mandelbaum et al,’ is presented in Fig. 3. The authors have estimated that the most abundant ions in this plasma are Pr M’+ . They have also found, by means of detailed line-by-line calculations in ions of several rare-earth elements, that configuration mixing is an important phenomenon in the N electronic shell (principal quantum number n = 4). For interpreting this spectrum in more detail, we run the code for each of the four ions considered. In each case, the configurations considered are: (i) the ground configuration C,, which is of the 4s24p64dk type; (ii) all the configurations deduced from C, by the excitation of one or two electrons within the N shell, i.e., 4/ -+ 4/’ excitations; (iii) 10 low configurations deduced from the latter by 4f + 5p or 4f+ 6p single replacements. These configurations are introduced because they also generate some 4&-4e’ transition arrays, whose average wavelengths lie in the range of the experimental spectrum. For each of the ions Pr’s-‘7+ , this sums up to 28 configurations, which generate 50 arrays. For Pr 14+, there are only 25 configurations, and 41 arrays, because the ground configuration is 4dg. A number of array strengths computed without and with the mixing effects are presented in Fig. 4(a) and (b), respectively, in the form of vertical sticks proportional to their value. They are
Evaluation
of configuration-interaction
47
effects
Praseodymium
Fig. 3. Emission
90
85
80
spectrum
of a Praseodymium
plasma
between
95
75 and 95 A (Ref. 2).
the strengths of all the arrays emitted by singly-excited configurations, whereas those emitted by doubly- or triply-excited configurations have been discarded, for a reason discussed below. Two conspicuous effects appear. First, there is a general weakening of all three types of 4e-W arrays: of course, the corresponding strength is transferred to the discarded arrays, because the total strength is not changed by the mixing. Secondly, the bunch of 4p-4d arrays, which lies at 1 = 100 + 5 I$, nearly vanishes. A more subtle effect is the shift of the center of gravity of the 4d4f arrays from about 1 = 84 to 81 A. For the comparison with the experimental spectrum, one must evaluate the populations of the emitting configurations. The plasmas generated by high-voltage vacuum sparks are short-lived, and probably farther from the Local Thermodynamic Equilibrium (LTE) than from the Coronal Equilibrium (CE). However, for the sake of simplicity, we present in Fig. 5(a) and (b) spectra simulated in LTE. They are derived from the strengths computed for all the 191 arrays of the four ions. It is assumed that the ions have equal spatial densities, but the occurrence of another ionic balance would not change the plot appreciably. Each unresolved transition array (UTA) is represented with a Gaussian profile, whose width is equal to one fourth of its width in the absence of mixing. This simple rule is used in analogy with the quantitative result obtained in a case of “quenching”, i.e., of vanishing of an array through configuration mixing.’ This analogy is valid, because the arrays lying around ,J = 81 8, have “quenched” the 4p-4d arrays. The temperatures are 50 and 150 eV in Fig. 5(a) and (b), respectively. The agreement between the peaks of Fig. 5(a) and of the experimental spectrum (Fig. 3) is good (the 3 %,difference between the average wavelengths may be attributed to the overestimation of the ab-initio exchange integrals).’ The shoulder lying to the high-wavelength side of the peak is essentially emitted by the 4p4d and 4d-4f arrays which are discarded from Fig. 4 because they are emitted by doubly- or triply-excited configurations. A confirmation that the levels of these configurations have small populations is found in Fig. 5(b), whose shoulder is more intense, only because the temperature increase favors the population of the high levels. In conclusion: (i) the 4p-4d low arrays are almost totally erased through their mixing with the 4d4f low arrays; (ii) the strengths of the 4d-4f low arrays are drastically reduced through their mixing with strong arrays generated by high, doubly-excited, highly degenerate, broad, upper configurations;
{ no mlxlng I
Pr xv-XVI-xvII-xvIII
1000.0 -
900.0
-
600.0 -
700.0 -
600.0 j
Ip-4d
500.0 400.0
4A- 4p 300.0 i 200.0
u+-.lk
,O 105.0 115.0 125.0 135.0 145.0 155.0 165.0 175.0
{ with mlxlng 1
Pr XV-XVI-XVII-XVIII
1000.0 -
900.0
-
600.0 -
700.0 -
600.0 -
500.0400.0300.0200.0lOO.O0.0 I I I I , I I I I , m'vI.,, I I ,,,,~,,~~,.;:~~~,,~,,,,,,,,,.,,,,,!:.',,,,!;,,.,, 45.0 55.0 65.0 75.0 65.0 95.0 105.0 115.0 125.0 136.0 145.0 155.0 165.0 175.0 h (A, Fig. 4. The computed array strengths in the spectra of Pr XV through XVIII. Each array is represented by a vertical segment, whose length is proportional to its strength. Actually, 191 arrays are introduced in the computation, but those which are emitted by multiply-excited configurations are not represented (see Sec. 5). (a) Without configuration interaction. (b) With configuration interaction. 48
Evaluation
of configuration-interaction
49
effects
(iii) from the experimental spectrum, one deduces that the populations of the levels of these upper configurations are very small; were it possible to assign a temperature to the plasma, it would be lower than 50 eV. 6. COMPUTATION
OF THE SECOND-ORDER CHANGES STRENGTHS OF THE ARRAYS
IN THE TOTAL
The second-order mixing generates new so-called “forbidden” arrays, between configurations of opposite parity, but which are not linked by electricdipolar lines when they are pure. For example, the levels of the configuration C, = d4p are linked by electricdipolar lines with the levels of C3 = d5, but those of C, = d3sp are not. However, if the latter levels are mixed with those of d4p, they can have electricdipolar lines towards d5. Pr XV-XVIII
0.0: . 75.0
8 .
.
,
00.0
-
T=SO ev
*
1
1
I wlth mlxlng 1
,
’
7
1
.
85.0
h
1
.
’
*
.
90.0
1 85.0
6)
I wlth mlxlng I
Pr xv-Jo/Ill T-150 ev 1000.0 900.0 800.0 700.0 600.0 500.0 400.0 300.0 zoo:0 100.0
9
0.0, 76.0
.
.
.
, 80.0
.
.
I
.
, 85.0
.
.
I
.
,
1
.
.
.
90.0
h A
Fig. 5. The computed emission spectrum of all the 191 unresolved arrays in the spectra of Pr XV through XVIII. Configuration interaction is taken into account. Each array is represented by a Gaussian curve. The vertical scale is in arbitrary units in (a) and (b). The plasma is supposed to be in LTE at temperature T. (a) T = 50 eV. (b) T = 150 eV.
( 05.0
50
J. Bauche
et al
Table 1. Second order configuration-interaction effects. The upper three lines of the two right columns are deduced from one diagonalization of the d4p + d”sp ensemble, and the lower four from one diagonalization of d4p + d2sZp.The forbidden strength is tabulated as a percentage of the allowed strength. The quantities between brackets are the rounded-off numbers of lines in the arrays allowed array
forbidden array
forbidden strength (W
d4p-d5
d3sp.
(3290)
(3970)
d3sp
- d2sp2
(22800)
d5
d4p - d2sp2
d4p - d3s2
(2100)
(1740)
d4p-d5
d%2p-d5
(3290)
(330)
d‘lp - d3p2
d2s2p - d3p2
(22600)
(5990)
d2s2p-d3s2
d4p
(470)
(1740)
(740)
0.44
0.41
0.40
0.41
0:43
0.066
0.065
0.066
0.066
0.31
0.31
0.31
0.31
(18900)
d3sp - d3s2
d%2p
0.47
_ d&2
- d3s2
d4p - ds2p2 (2620)
Let us suppose that the mixing configurations lie far apart one from the other. Then, to the first order of perturbation by the G operator, the wavefunction $2 of a level of the C, configuration reads where the square brackets mean that the enclosed state is impure, and where E(C,) and E(C,) are the average energies of the states of the respective configurations. The sum runs over all the states of the C, configuration. Therefore, the total relative admixture of C, into C, is where g, is the degeneracy of configuration C, . Because the off-diagonal energy element (C, 11/,IGlC,tj,) and the amplitudes a(C, $, , C,I++~)of the transitions towards the states of another configuration C, are clearly uncorrelated, the T(C, --+ C,) admixture and the C,-C, total transition strength are, on the average, proportional. The latter assertion can be tested numerically by means of detailed line-by-line calculations. Such tests are presented in Table 1 for seven forbidden transition arrays. For the calculations, the values of the radial parameters have been chosen somewhat arbitrarily, with the condition that the energy difference between the mixing configurations is large with respect to their energy widths. The agreement between the computed admixtures and strengths is excellent. The important point is that the double sum Zti,92(C,$,IGIC,$,)2 in the numerator of Eq. (5) can be computed using compact formulas. This makes also the total forbidden strength expandable as a compact formula, in a very good approximation. 7.
CONCLUSION
A computer code has been built, which yields the effects of configuration interaction on the total strengths of the electric-dipolar transition arrays. Using this code, the effects observed in Pr XV through Pr XVIII, which are of an exceptional magnitude, are interpreted quantitatively.
Evaluation
of configuration-interaction
effects
51
Several works are in progress for improving the model. For deducing the corresponding emission or absorption spectrum, the changes in the shape and in the width of each array must be studied. If, before the mixing procedure, an array is split into subarrays by some large spinorbit interaction effects, another model is needed for computing the interaction between sub-arrays belonging to distinct arrays. In very heavy highly-ionized atoms, the case of the interaction between a whole array and a sub-array of another array has been observed. It should also be possible to compute this effect in a global way. A general code is being built for computing the second-order mixing effects in any spectrum. It will give the total strengths of the so-called forbidden transition arrays, whether they result from second- or higher-order effects. An essential question is to be solved: what happens when the average energies of two configurations, which are subject to second-order mixing, are nearly identical? Acknowledgement-Work
performed
under the auspices of the U.S. Department
of Energy, Contract
No. W-7405-ENG-48.
REFERENCES 1. J. Bauche, C. Bauche-Arnoult, and M. Klapisch, Adu. Atom. Mofec. Phys. 23, 131 (1987). 2. P. Mandelbaum, M. Finkenthal, J.-L. Schwab, and M. Klapisch, Phys. Reu. A 35, 5051 (1987). 3. J. Bauche, C. Bauche-Arnoult, M. Klapisch, P. Mandelbaum, and J.-L. Schwab, J. Phys. B, Atom. Molec. Phys. 20, 1443 (1987).
4. 5. 6. 7. 8.
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press (1935). J. Bauche, C. Bauche-Arnoult, and M. Klapisch, J. Phys. B, Atom. Molec. Phys, 24, I (1991). J. Bauche, C. Bauche-Arnoult, E. Luc-Koenig, J.-F. Wyart, and M. Klapisch, Phys. Rev. A 28,829 (1983). J. Bauche and C. Bauche-Arnoult, J. Phys. B, Atom. Molec. Phys. 22, 2503 (1989). E. Koenig, Physica 62, 393 (1972); M. Klapisch, J.-L. Schwab, B. S. Fraenkel, and J. Oreg, J. opt. Sot. Am. 67, 148 (1977).
Pergamon
Specwosc. Radiar.
0022-4073(95)00039-9
Transfer Vol. 54, No. l/2. pp. 43-51, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-4073195 $9.50 + 0.00
EVALUATION OF CONFIGURATION-INTERACTION EFFECTS ON ATOMIC TRANSITION ARRAYS J. BAUCHE,_F C. BAUCHE-ARNOULT,?
A. BACHELIER,?
and
W. H. GOLDSTEIN1 tLaboratoire
Aimt
Cotton,
CNRS II, 91405 Orsay, France and iLawrence Laboratory, Livermore, CA 94551, U.S.A.
Livermore
National
Abstract-In the field of the ab-initio simulations of atomic spectra, a method is described for taking into account the configuration-interaction effects on electric-dipolar transition arrays. As an application, the strong effects observed in the 4f-4d’ arrays of the spectra of Praseodymium XV through XVIII are interpreted quantitatively. An original method is proposed for computing the intensities of the “forbidden” electric-dipolar arrays, which are generated by configuration interaction.
1. INTRODUCTION
We report here current results obtained in the evaluation of global effects of configuration interaction in atomic spectra. Global means that the effects studied relate to groups of levels (the electronic configurations) and of radiative lines (the transition arrays), but not to individual levels or lines.’ If configuration interaction is considered as a small perturbation of the central-field scheme, its effects on energy levels and line intensities are, at most, of the second and first orders, respectively. Dramatic first-order changes in the intensities of some transition arrays have already been observed, and interpreted. 2,3In the present work, that method of interpretation is generalized to more complex situations, and results from a computer code are presented. Another method is proposed for the case where the intensities are only perturbed to the second order. In the following, the words interaction and mixing are equivalent, because the perturbations studied are actually generated by the mixing of the quantum states of different configurations. The configurations considered are those defined by Condon and Shortley.4 They are not the relativistic (sub-) configurations whose mixing has been treated elsewhere.5
2. WHAT
IS KNOWN
UP TO NOW
Let us consider three electronic configurations C,, C,, C,, linked by the electric&polar transition arrays C,--C, and C,-C,, denoted C,-(C, + C,) altogether (Fig. 1). If the levels of configurations C, and C, are mixed through the electrostatic interelectronic-repulsion operator G, the intensities of the radiative lines are changed. In the following, we are essentially interested in the quantum part of the line intensities, i.e., in the line strengths, and in the total array strengths, which are the sums of the corresponding line strengths.6 The strength of a line is the square of its amplitude, and can be written, in tensor-operator form
for the line between the levels a,J, and azJ2. The LXsymbols refer to the intermediate couplings in the relevant configurations. D(I) is the electric-dipolar tensor operator. For the effects of configuration mixing on the array strengths, there appear two different situations. 43
J. Bauche et al
44
2.1. First -order array mixing
What we call the first-order mixing can be characterized by two properties, which are equivalent. (i) The energy W of the center of gravity of the energies of all the lines of both arrays, weighted by their strengths, is changed by the mixing. (ii) If the C, and C, configurations lie far apart one from the other, the changes of the line strengths, and of W, are linear functions of the Rk Slater integrals responsible for the mixing. Example: the dN + ,s_(dN + ’p + dNsp) mixing, in the transition-metal spectra. 2.2. Second-order array mixing In the case of a second-order array mixing, the properties (i)+ii) above are replaced by two other ones. (i) The center of gravity W is not changed by the mixing. (ii) If the C, and C, configurations lie far apart one from the other, the changes of the line strengths, and of W, are quadratic functions of the Rk Slater integrals responsible for the mixing. These properties are also valid for the cases where the electric-dipolar transition array between C3 and one of the other configurations is said to be “forbidden”, i.e. when it is generated by the mixing. Example: the dNs2-(dN+ ‘p + dNsp) mixing, in the transition-metal spectra. 2.3. Computation of the$rst-order
changes in the total strengths of the arrays
A matrix method has been proposed a few years ago for the computation of the first-order changes in the array strengths.’ The matrix in Fig. 1, corresponding to the C3-(C, + C,) case, is called a triad. In the energy matrix, the diagonal elements E, , E, and E, are the weighted average energies of the levels of the configurations C, , C, and C,, respectively, in the absence of mixing. The weight of each level is its emissivity, defined as the sum of the strengths of all the lines related to this level. E,, E2 and E3 are also called the average energies of the emissive zones.6 The off-diagonal elements a,, and a23 are the total amplitudes of the arrays C,--C, and C,-C,, respectively, defined as the positive square roots of their total strengths. All these quantities can be computed using compact formulas.’ The off-diagonal energy matrix element El2 is defined in the following way. When the energy matrix is diagonalized, with eigenvalues denoted E; and E;, its eigenvectors (a, /3) and (j?, -a) are used for obtaining the transformed strengths s;~ = (u;~)~ and s& = (a;,)2, with ai = UZ,~+ flu23and a& = Ba,3 - ua23. Thus, in the occurrence of the mixing, the value of the weighted center of gravity of the array energies reads W’=
[s;jE; +s;,E;]/[s;J
+s;J
- E3.
(2)
The general formal expression of E,, has been derived in order that W’ be equal to the weighted center of gravity of the line energies, as obtained through a detailed line-by-line calculation.’ Using this expression also ensures that the transformed strengths are exact to the first order with respect to the Rk Slater integrals, i.e. when these integrals are infinitely small.
3. A MORE
GENERAL
METHOD
FOR
COMPUTING
THE
FIRST-ORDER
MIXING
The matrix in Fig. 2 corresponds to the case denoted (C, + C,) - (C, + C,), called a tetrad, where two even configurations, C, and C,, are both linked by electricdipolar transitions with two odd ones, C, and C,. This case is more complicated than that of a triad, but it is not rare in the spectra. The energy and amplitude matrix elements are computed as it is recalled in Sec. 2.3.
of configuration-interaction
effects
I
I E,
Cl
1
Et2
1
Q13
Fig. 1. Matrix for computing the case of the triad C,
After all four energy matrices are diagonalized, the transformed strengths si3 through siq are exact to the first order with respect to the Rk integrals. However, in general, their sum is not rigorously equal to &, =
s13 +
s23 +
sl4
+
(3)
s24
the value before the diagonalization, in contrast with the detailed line-by-line method. This is due to the fact that, in the matrix method used, the occurrence of higher-order contributions is not taken into account. Furthermore, all the elements of the 4 x 4 off-diagonal amplitude matrix in Fig. 2 are now different from zero.
c2
Fig. 2. Matrix for computing the case of the tetrad (C, + C&o, + Cz). A tetrad is the combination of 4 intermingled triads. This is the most general case, where each configuration has one emissive zone relating to each configuration of the other parity. For example, E, (resp. H,) is the energy of the C, emissive zone relating to the emission towards C3 (resp. C,). In this figure, the energy order of the 8 emissive zones is arbitrary.
J. Bauche et al
46
Two approximate
methods are proposed for correcting this drawback. (i) In the first method, one simply renormalizes the strengths in order that their sum be equal to &0,. (ii) The second method uses the eigenvectors in a different way. In the first step, one transforms the initial amplitudes by means of the eigenvectors of the upper two (odd) matrices in Fig. 2; then, one transforms the results by means of the eigenvectors of the even matrices. This yields a set of four strengths whose sum is equal to S,,,. The second step is identical to the first one, but for the order in which the odd and even eigenvectors are used. The final values of the strengths are the half-sums of the results of the two steps. It can be proved that any spectrum is the association of many intermingled triads and tetrads (no more complicated case can occur), if one discards the mixing between configurations differing only by the principal quantum number n of one orbital. The latter simplification is valid for the HartreeFock radial functions, which are known to be fairly close to the central-field parametricpotential functions. In each case, even and odd matrices are built like those in Fig. 2. Each transition array relates to one even and one odd basis vector. Therefore, the orders of the whole even and odd matrices are both equal to the number of arrays. 4. THE
CODE
FOR
COMPUTING
FIRST-ORDER
MIXING
EFFECTS
A computer code has been written for computing the first-order mixing effects on the array strengths of a given spectrum. The input data file contains the list of the configurations to be considered. All the corresponding electricdipolar transition arrays are automatically taken into account in the calculation. The first step is the use of the ab-initio relativistic parametric-potential code RELAC8 which generates all the needed radial integrals, and deduces the average energies of the configurations, and of their emissive zones. In the second step, the even and odd configurations are implemented, and diagonalized. No large-scale diagonalization is required, because both matrices break into many block-diagonal smaller matrices. In Fig. 2, both 4 x 4 energy matrices break into two 2 x 2 submatrices. In the Pr calculations (Section 5), the largest submatrix which we have encountered was 7 x 7. Then, the eigenvectors are used for computing the transformed strengths, in the two ways described in Section 3. The size limitations of the code are essentially those of RELAC, for the maximum number of configurations which can be treated. Otherwise, the number of arrays is not limited. 5. EXAMPLE:
THE
IONS
PrXV
THROUGH
PrXVIII
A part of the experimental praseodymium spectrum generated by a high-voltage vacuum spark, published by Mandelbaum et al,’ is presented in Fig. 3. The authors have estimated that the most abundant ions in this plasma are Pr M’+ . They have also found, by means of detailed line-by-line calculations in ions of several rare-earth elements, that configuration mixing is an important phenomenon in the N electronic shell (principal quantum number n = 4). For interpreting this spectrum in more detail, we run the code for each of the four ions considered. In each case, the configurations considered are: (i) the ground configuration C,, which is of the 4s24p64dk type; (ii) all the configurations deduced from C, by the excitation of one or two electrons within the N shell, i.e., 4/ -+ 4/’ excitations; (iii) 10 low configurations deduced from the latter by 4f + 5p or 4f+ 6p single replacements. These configurations are introduced because they also generate some 4&-4e’ transition arrays, whose average wavelengths lie in the range of the experimental spectrum. For each of the ions Pr’s-‘7+ , this sums up to 28 configurations, which generate 50 arrays. For Pr 14+, there are only 25 configurations, and 41 arrays, because the ground configuration is 4dg. A number of array strengths computed without and with the mixing effects are presented in Fig. 4(a) and (b), respectively, in the form of vertical sticks proportional to their value. They are
Evaluation
of configuration-interaction
47
effects
Praseodymium
Fig. 3. Emission
90
85
80
spectrum
of a Praseodymium
plasma
between
95
75 and 95 A (Ref. 2).
the strengths of all the arrays emitted by singly-excited configurations, whereas those emitted by doubly- or triply-excited configurations have been discarded, for a reason discussed below. Two conspicuous effects appear. First, there is a general weakening of all three types of 4e-W arrays: of course, the corresponding strength is transferred to the discarded arrays, because the total strength is not changed by the mixing. Secondly, the bunch of 4p-4d arrays, which lies at 1 = 100 + 5 I$, nearly vanishes. A more subtle effect is the shift of the center of gravity of the 4d4f arrays from about 1 = 84 to 81 A. For the comparison with the experimental spectrum, one must evaluate the populations of the emitting configurations. The plasmas generated by high-voltage vacuum sparks are short-lived, and probably farther from the Local Thermodynamic Equilibrium (LTE) than from the Coronal Equilibrium (CE). However, for the sake of simplicity, we present in Fig. 5(a) and (b) spectra simulated in LTE. They are derived from the strengths computed for all the 191 arrays of the four ions. It is assumed that the ions have equal spatial densities, but the occurrence of another ionic balance would not change the plot appreciably. Each unresolved transition array (UTA) is represented with a Gaussian profile, whose width is equal to one fourth of its width in the absence of mixing. This simple rule is used in analogy with the quantitative result obtained in a case of “quenching”, i.e., of vanishing of an array through configuration mixing.’ This analogy is valid, because the arrays lying around ,J = 81 8, have “quenched” the 4p-4d arrays. The temperatures are 50 and 150 eV in Fig. 5(a) and (b), respectively. The agreement between the peaks of Fig. 5(a) and of the experimental spectrum (Fig. 3) is good (the 3 %,difference between the average wavelengths may be attributed to the overestimation of the ab-initio exchange integrals).’ The shoulder lying to the high-wavelength side of the peak is essentially emitted by the 4p4d and 4d-4f arrays which are discarded from Fig. 4 because they are emitted by doubly- or triply-excited configurations. A confirmation that the levels of these configurations have small populations is found in Fig. 5(b), whose shoulder is more intense, only because the temperature increase favors the population of the high levels. In conclusion: (i) the 4p-4d low arrays are almost totally erased through their mixing with the 4d4f low arrays; (ii) the strengths of the 4d-4f low arrays are drastically reduced through their mixing with strong arrays generated by high, doubly-excited, highly degenerate, broad, upper configurations;
{ no mlxlng I
Pr xv-XVI-xvII-xvIII
1000.0 -
900.0
-
600.0 -
700.0 -
600.0 j
Ip-4d
500.0 400.0
4A- 4p 300.0 i 200.0
u+-.lk
,O 105.0 115.0 125.0 135.0 145.0 155.0 165.0 175.0
{ with mlxlng 1
Pr XV-XVI-XVII-XVIII
1000.0 -
900.0
-
600.0 -
700.0 -
600.0 -
500.0400.0300.0200.0lOO.O0.0 I I I I , I I I I , m'vI.,, I I ,,,,~,,~~,.;:~~~,,~,,,,,,,,,.,,,,,!:.',,,,!;,,.,, 45.0 55.0 65.0 75.0 65.0 95.0 105.0 115.0 125.0 136.0 145.0 155.0 165.0 175.0 h (A, Fig. 4. The computed array strengths in the spectra of Pr XV through XVIII. Each array is represented by a vertical segment, whose length is proportional to its strength. Actually, 191 arrays are introduced in the computation, but those which are emitted by multiply-excited configurations are not represented (see Sec. 5). (a) Without configuration interaction. (b) With configuration interaction. 48
Evaluation
of configuration-interaction
49
effects
(iii) from the experimental spectrum, one deduces that the populations of the levels of these upper configurations are very small; were it possible to assign a temperature to the plasma, it would be lower than 50 eV. 6. COMPUTATION
OF THE SECOND-ORDER CHANGES STRENGTHS OF THE ARRAYS
IN THE TOTAL
The second-order mixing generates new so-called “forbidden” arrays, between configurations of opposite parity, but which are not linked by electricdipolar lines when they are pure. For example, the levels of the configuration C, = d4p are linked by electricdipolar lines with the levels of C3 = d5, but those of C, = d3sp are not. However, if the latter levels are mixed with those of d4p, they can have electricdipolar lines towards d5. Pr XV-XVIII
0.0: . 75.0
8 .
.
,
00.0
-
T=SO ev
*
1
1
I wlth mlxlng 1
,
’
7
1
.
85.0
h
1
.
’
*
.
90.0
1 85.0
6)
I wlth mlxlng I
Pr xv-Jo/Ill T-150 ev 1000.0 900.0 800.0 700.0 600.0 500.0 400.0 300.0 zoo:0 100.0
9
0.0, 76.0
.
.
.
, 80.0
.
.
I
.
, 85.0
.
.
I
.
,
1
.
.
.
90.0
h A
Fig. 5. The computed emission spectrum of all the 191 unresolved arrays in the spectra of Pr XV through XVIII. Configuration interaction is taken into account. Each array is represented by a Gaussian curve. The vertical scale is in arbitrary units in (a) and (b). The plasma is supposed to be in LTE at temperature T. (a) T = 50 eV. (b) T = 150 eV.
( 05.0
50
J. Bauche
et al
Table 1. Second order configuration-interaction effects. The upper three lines of the two right columns are deduced from one diagonalization of the d4p + d”sp ensemble, and the lower four from one diagonalization of d4p + d2sZp.The forbidden strength is tabulated as a percentage of the allowed strength. The quantities between brackets are the rounded-off numbers of lines in the arrays allowed array
forbidden array
forbidden strength (W
d4p-d5
d3sp.
(3290)
(3970)
d3sp
- d2sp2
(22800)
d5
d4p - d2sp2
d4p - d3s2
(2100)
(1740)
d4p-d5
d%2p-d5
(3290)
(330)
d‘lp - d3p2
d2s2p - d3p2
(22600)
(5990)
d2s2p-d3s2
d4p
(470)
(1740)
(740)
0.44
0.41
0.40
0.41
0:43
0.066
0.065
0.066
0.066
0.31
0.31
0.31
0.31
(18900)
d3sp - d3s2
d%2p
0.47
_ d&2
- d3s2
d4p - ds2p2 (2620)
Let us suppose that the mixing configurations lie far apart one from the other. Then, to the first order of perturbation by the G operator, the wavefunction $2 of a level of the C, configuration reads where the square brackets mean that the enclosed state is impure, and where E(C,) and E(C,) are the average energies of the states of the respective configurations. The sum runs over all the states of the C, configuration. Therefore, the total relative admixture of C, into C, is where g, is the degeneracy of configuration C, . Because the off-diagonal energy element (C, 11/,IGlC,tj,) and the amplitudes a(C, $, , C,I++~)of the transitions towards the states of another configuration C, are clearly uncorrelated, the T(C, --+ C,) admixture and the C,-C, total transition strength are, on the average, proportional. The latter assertion can be tested numerically by means of detailed line-by-line calculations. Such tests are presented in Table 1 for seven forbidden transition arrays. For the calculations, the values of the radial parameters have been chosen somewhat arbitrarily, with the condition that the energy difference between the mixing configurations is large with respect to their energy widths. The agreement between the computed admixtures and strengths is excellent. The important point is that the double sum Zti,92(C,$,IGIC,$,)2 in the numerator of Eq. (5) can be computed using compact formulas. This makes also the total forbidden strength expandable as a compact formula, in a very good approximation. 7.
CONCLUSION
A computer code has been built, which yields the effects of configuration interaction on the total strengths of the electric-dipolar transition arrays. Using this code, the effects observed in Pr XV through Pr XVIII, which are of an exceptional magnitude, are interpreted quantitatively.
Evaluation
of configuration-interaction
effects
51
Several works are in progress for improving the model. For deducing the corresponding emission or absorption spectrum, the changes in the shape and in the width of each array must be studied. If, before the mixing procedure, an array is split into subarrays by some large spinorbit interaction effects, another model is needed for computing the interaction between sub-arrays belonging to distinct arrays. In very heavy highly-ionized atoms, the case of the interaction between a whole array and a sub-array of another array has been observed. It should also be possible to compute this effect in a global way. A general code is being built for computing the second-order mixing effects in any spectrum. It will give the total strengths of the so-called forbidden transition arrays, whether they result from second- or higher-order effects. An essential question is to be solved: what happens when the average energies of two configurations, which are subject to second-order mixing, are nearly identical? Acknowledgement-Work
performed
under the auspices of the U.S. Department
of Energy, Contract
No. W-7405-ENG-48.
REFERENCES 1. J. Bauche, C. Bauche-Arnoult, and M. Klapisch, Adu. Atom. Mofec. Phys. 23, 131 (1987). 2. P. Mandelbaum, M. Finkenthal, J.-L. Schwab, and M. Klapisch, Phys. Reu. A 35, 5051 (1987). 3. J. Bauche, C. Bauche-Arnoult, M. Klapisch, P. Mandelbaum, and J.-L. Schwab, J. Phys. B, Atom. Molec. Phys. 20, 1443 (1987).
4. 5. 6. 7. 8.
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press (1935). J. Bauche, C. Bauche-Arnoult, and M. Klapisch, J. Phys. B, Atom. Molec. Phys, 24, I (1991). J. Bauche, C. Bauche-Arnoult, E. Luc-Koenig, J.-F. Wyart, and M. Klapisch, Phys. Rev. A 28,829 (1983). J. Bauche and C. Bauche-Arnoult, J. Phys. B, Atom. Molec. Phys. 22, 2503 (1989). E. Koenig, Physica 62, 393 (1972); M. Klapisch, J.-L. Schwab, B. S. Fraenkel, and J. Oreg, J. opt. Sot. Am. 67, 148 (1977).