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Oscillator strength ratios for the principal series of rubidium : relativistic and correlation effects
E. Luc-Koenig, A. Bachelier
To cite this version:
E. Luc-Koenig, A. Bachelier. Oscillator strength ratios for the principal series of rubid- ium : relativistic and correlation effects. Journal de Physique, 1978, 39 (10), pp.1059-1063.
�10.1051/jphys:0197800390100105900�. �jpa-00208845�
1059
OSCILLATOR STRENGTH RATIOS FOR THE PRINCIPAL SERIES
OF RUBIDIUM : RELATIVISTIC AND CORRELATION EFFECTS
E. LUC-KOENIG and A. BACHELIER
Laboratoire Aimé-Cotton, C.N.R.S. II, Bât. 505, 91405 Orsay, France (Reçu le 12 mai 1978, accepté le 4 juillet 1978)
Résumé. 2014 Les forces d’oscillateur f3/2n et f1/2n de la série principale du rubidium sont calculées à
partir des fonctions d’onde relativistes données par la méthode du potentiel paramétrique. Trois
études différentes permettent de montrer que la séparation entre effets de corrélation et effets rela- tivistes n’est pas unique et quelque peu arbitraire. Les calculs sont effectués pour des valeurs du nombre quantique principal atteignant n = 80 et montrent que le rapport Rn = f3/2n /f1/2n croît rapidement avec n et tend vers une valeur limite, conformément à une étude expérimentale récente portant sur les états de Rydberg de Rb.
Abstract. 2014 Oscillator strengths f3/2n and f1/2n for the principal series of rubidium are calculated
using relativistic wave functions obtained by means of the relativistic parametric-potential method.
Three different approaches are used pointing out that the separation between relativistic and cor-
relation effects is not unique and somewhat arbitrary. The calculations were carried out up to the
principal quantum number n = 80 showing that the ratio Rn = f3/2n / f1/2n increases rapidly with n
towards a limiting value, a result in agreement with a recent experiment on highly excited Rydberg
states.
LE JOURNAL DE PHYSIQUE TOME 39, OCTOBRE 1978,
Classification
Physics Abstracts
32.70
1. Introduction. - Recently anomalous intensities have been observed in the fine-structure patterns for the 5s-np transitions (29 n 50) of Rbl [1]. For
all patterns which have been measured the ratio Rn
of the intensities of the two components In3/2 and Inl/2 of the doublet differs from the ratio of the statis- tical weights of the n 2P3J2 and n 2P 1/2 excited levels : instead of R th = 2, the observed value is equal to
Rexp = 5.9 + 1.4 and, within the experimental uncer- tainty, does not depend on n. However no experimen-
tal data are available for the low lying levels, except
for n = 5 where R5exp = 1.99 ± 0.34 [2]. Therefore
there is no experimental information on the evolution of Rn between n = 6 and n = 28.
Similar anomalies were observed for the first time in 1930 for the resonance lines of caesium. They were explained qualitatively by Fermi [3] as resulting from
the strong non-diagonal spin-orbit interaction bet-
ween the n 2Pj levels. To first order in perturbation theory, the contributions to f3/2 and fn1/2 coming
from the excited levels m, such as Em > En, tend to
reduce Rn, but these contributions decrease rapidly with m ; on the other hand the perturbing levels m
with Em En correspond to a positive correction to
Rn. The perturbations coming from the low lying
levels are predominant, since these levels are associated with the strongest transition matrix elements. Thus
Rn increases with n and tends towards a limit value [4],
as the admixture with the lowest n ’Pj levels contri- butes the predominant part of the departure of Rn
from R th. Further theoretical or experimental inves- tigations of the ratio for high lying levels lead to diffe- rent predictions for the behaviour of Rn : from some
results Rn would pass through a maximum as n
increases [5-8], from the others the ratio would keep
an increasing [9-12]. No experimental data are avai-
lable for n > 19, and in the most recent experiments [9, 10, 13] no maximum was found for the ratio.
Several theoretical calculations of the oscillator
strengths for the principal series of alkali-metals have been performed using non relativistic wave-
functions obtained from semi-empirical model poten- tials introducing spin-orbit and core polarization
effects [8, 12, 14]. The matrix elements associated with transitions to highly excited levels are very sensitive to the part of the wavefunctions near the
nucleus, where the overlap of the 5s and np orbitals is not negligible. Consequently Norcross’s calculations
[12] give the more reliable results for CsI, since they introduce, besides the spin-orbit, additional relati- vistic corrections which are associated with the proper behaviour of the relativistic potential in the vicinity
of the nucleus.
The oscillator strengths for the principal series of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390100105900
rubidium have been calculated by Weisheit [8] from
a semi-empirical model based on a Hartree-Fock core
potential, which included explicitly the core polari-
zation and the spin-orbit interaction. The results are
reported in table 1 and in figure 1. The intensity ratios
obtained by Weisheit increase rapidly with n and, for
n = 17, the value of the ratio R W = 6.34 is near to
the experimental limit value R "P = 5.9 ± 1.4. Never-
TABLE 1
Intensity ratios for the resonance lines of rubidium
FIG. 1. - Intensity ratios for the resonance lines of rubidium.
Theory : w) Weisheit’calculation, Ref. [8].
a) semi-empirical calculation
b) variational calculation this work.
c) frozen-core calculation 1
Experiment : hatched area R exp = 5.9 ± 1.4, Ref. [1].
theless, Weisheit’s calculation was performed only for
n 17, and in this range it is not clear that the ratio should tend towards a limiting value. Moreover in
the case of caesium the use of the same approach
leads to a ratio passing through a maximum in the
vicinity of n = 16 ; the result does not agree with the most recent experimental results [9, 10] or with the
theoretical values obtained by Norcross [12]. Norcross
showed that the f value for the 5 2S1/2-n 2P 1/2 tran-
sition is strongly affected by the behaviour of the
spin-orbit interaction near the nucleus. Consequently
the reliability of the results obtained by Weisheit for the principal series of rubidium is questionable, since
Weisheit used for the spin-orbit interaction the expres-
sion -i OE 2 Z * 1.s/r3 which is not valid for small r. values.
The rather good agreement obtained by Weisheit
arises perhaps from the fact that the relativistic effects
are not as much predominant in rubidium as in
caesium, and that no large n values are considered.
In a previous study [15], one of us has shown that
it is possible to explain the line-strength anomalies
in the principal series of caesium, by a first order calculation of the relativistic-central-field approxima-
tion. In a subsequent paper [16], it was shown that
the inverted fine structures observed in alkali-like spectra are also related to purely relativistic effects, which modify the large components of the relativistic radial wavefunctions and which cannot be reproduced
in the Pauli-limit, if configuration interactions are
neglected. Moreover the relativistic corrections of order OE2 to the large components of the relativistic wavefunctions can simply be represented by a first
order calculation of configuration interactions through
the one-electron Pauli operators (Darwin, p4 term, spin-orbit) [17]. The anomalous Landé factors observ- ed experimentally in the sp configurations of Cd and Hg can likewise be reproduced in a first order calcu- lation of the relativistic central field approximation [18] ; in that example the relativistic corrections to the
operator obtained through the use of the Foldy- Wouthuysen transformation are almost negligible,
while the relativistic corrections to the wavefunc- tions are predominant.
In this paper, the study of the resonance lines for
the principal series of rubidium gives another example pointing out that the configuration interactions coming
from the spin-orbit operator are automatically intro-
duced in a first order calculation of the relativistic-
central-field ; moreover different approaches are used showing that relativistic and correlation effects are
closely related and that their respective importance depends upon the method used, so that their sepa- ration is somewhat arbitrary.
2. Relativistic calculations of the oscillator strengths
for the principal séries of rubidium. - The relativistic radial wavefunctions are obtained from the relativistic-
parametric-potential method [15] and the velocity for-
mulation for the oscillator strengths is used.
1061
We have used the velocity form of the transition operator since this operator samples the wavefunctions at relatively small r values. Indeed the different appro- aches used in sections 2. 2. 3 lead to more accurate determination of the np excited wavefunctions near
the nucleus, where the non local exchange potential arising from the core wavefunctions is not vanishing.
Moreover in the numerical calculation ôf the transition matrix elements the contribution of the 5s wave-
function is negligible beyond the 12th node of the excited np wavefunction (n > 15) so that the transi-
tion matrix elements do not allow to test the excited wavefunctions at large r values.
2.1 SEMI-EMPIRICAL CALCULATION FROM A MODEL- POTENTIAL. - The central potential is represented by
an analytic function which depends on three para- meters. The long-range core polarization effects are
introduced in an effective way by adding to the central
potential the correction
where ad is the static dipole polarizability of the core and r,,, an effective core radius ; for re and ad we use the values given by Weisheit [8]. The optimal potential
is obtained by fitting the eigenvalues of the Dirac
equation to the experimental energies of the corres- ponding levels. The 40 lowest odd and even levels of the spectrum are interpreted with a root mean square deviation equal to 14.1 cm-1. In this approach the
core wavefunctions do not appear in the calculation of the energy levels and the central potential intro-
duces in an effective way and on an average all confi-
guration interactions corresponding to an excitation
of the valence electron. Consequently the calculation of the oscillator strengths is meaningful only to the
first order of perturbation theory and the configuration
interaction must not be introduced explicitly. The
values obtained for Rn are given in the first column of table I. Rn increases rapidly with n and reaches a
limiting value ; for n greater than 20, the variation of
Rn does not exceed 0.2 %. The limiting value, which
is equal to 3.8, is smaller than the experimental result,
but is qualitatively in good agreement with the expe- rimental data. Even though this calculation is not
sophisticated it points out that a calculation to the first order of the relativistic central field, automatically
introduces configuration interaction coming from the
one-electron Pauli operators.
2.2 VARIATIONAL CALCULATION. - T 0 take into account core relaxation effects two different potentials
are used, one for the ground and one for excited levels.
The first minimizes the total first order energy of the
ground 5 2S 1/2 level. The second minimizes the total first order energy of an excited n 2Pj level. We have
chosen the 10 2p 3/2 level which is sufficiently excited
to represent the effect of high lying Rydberg p states ;
moreover for the 10 ’P3/2 level, the total energy
(- 2 979.613 5 H) differs significantly from the total energy of the ground state of the ion ( - 2 979.178 3 H),
so that a variational calculation is meaningful (the
relative deviation is equal to 2 x 10- 5): Similar
results are obtained from the minimization of the total energy of the 10 2p 1/2 level, and are not given
here. The core polarization term AVp is introduced
in the potential for the Rydberg p states, because
AVP is expected to make a significant contribution
only well outside the core. Moreover the overlap
of the core wavefunctions with the 5s orbital is important so that the expression for the polarization potential AV, is not valid in the study of the ground
state, since the cut-off function { 1 - e - (r/rc)6 } is
somewhat arbitrary.
The core wavefunctions are not the same in the
ground and in the excited levels, so that the overlap integrals (nlj n’lj) are no longer equal to bnn, ; never-
theless the off-diagonal overlap integrals (n #- n’) are generally small. Consequently, to the first order of perturbation theory, the square root of the oscillator
strengths is written as a sum of several terms ; the
corresponding expression can be arranged in ascending
order of the number of non-diagonal overlap integrals.
We thus let Rn(p) denote the calculated oscillator
strengths that take into account the terms which contain the product of at most p non-diagonal inte- grals.
The terms which introduce the product of q non- diagonal integrals represent approximately the contri-
bution of the excitations of q electrons of the core, in the approach which introduces a single central potential for the ground and excited states.
For the variational calculation, Rn(0) and Rn(1) are
not very different, as can be seen in table II. The rela-
tive difference between Rn(0) and Rn(1) is not greater than 2 % and increases with n, pointing out that the
transitions to high lying Rydberg states are the most
sensitive to correlation effects. Core and valence orbitals occur simultaneously in the optimization procedure, so that the central potential introduces partially, and in an effective way, correlation effects into the wavefunctions.
The contribution of the core polarization potential
cannot be neglected, especially for large n values ;
the values obtained without introducing AVP in the
TABLE II
Variational calculation of the intensity ratios for the
resonance lines of rubidium
1062
calculation of the Rydberg states are given in column 3
of table II. For n = 30, the value 3. Il obtained without A Vp, is intermediate between R lh = 2 and
R30(2) = 4.64. Moreover, for the oscillator strengths,
f103/2 and f l02 increase by a factor of respectively 2.5
and 3.4, when A Vp is neglected.
The values obtained for Rn(1) are reported in the second column of table I. Rn(1) increases more rapidly
with n than in the semi-empirical calculation. More-
over the limiting value 4.7 is greater than in the pre-
ceding approach and does not vary significantly for
20 n 80. Experimental and theoretical results
are in good agreement pointing out that core relaxa-
tion effects cannot be neglected.
2. 3 FROZEN - CORE APPROXIMATION. - In this
approach the 5s ground state wavefunction is kept
the same as in the preceding section. In order to obtain
a more exact description of high lying Rydberg states, it is possible to use for the core orbitals the wave-
functions of the ground state of the positive ion.
Indeed, in the study of the fine structure inversions in the alkali-metal spectra [16] we have adapted the relativistic-parametric-potential method to the study
of Rydberg states : the latter treatment gives better
results that those obtained from the semi-empirical
calculation. The core wavefunctions for the Rydberg
states were obtained by minimizing the total energy of the ground level of the positive ion ; the valence electron wavefunction was calculated in the central
potential VH resulting from the nucleus and the direct part of the electrostatic interaction with the core
electrons. In the particular case of sodium nd and cesium nf Rydberg states, the orthogonality condi-
tions between core and valence electrons were auto-
matically satisfied, because these orbitals correspond
to different 1 values.
In the present work two different methods can be utilized to obtain the n 2Pj wavefunctions. In the first
they are calculated in the central direct potential VH ; then they are orthogonalized with respect to the core wavefunctions of the same symmetry. In the second,
the non-local exchange potential is added and the
orthogonality of wavefunctions is ensured by the
introduction of La’.grange multipliers. An iterative pro- cedure is used to solve the systems of inhomogeneous,
first order differential coupled equations. The core polarization potential is introduced in the calculation.
In table III we present the numerical results obtained for Rn, n = 5, 10, 30 and 50, using different approxi-
mations to calculate the n 2Pj wavefunctions, and introducing the p first terms in the evaluation of the oscillator strengths. In the first row the wavefunctions
are calculated with the direct potential VH, and only
the first term Rn(0) is kept ; this approach is insuffi- cient to reproduce the experimental data, since R30(0) = 2.33 is not very different from the classical value R th = 2. In the second row orthogonalization
is introduced afterwards, but the corresponding cor-
TABLE III Frozen-core approximation
rection is almost negligible ; moreover the discrepancy
between Rn(0) and Rn(1) does not exceed 1 %. These
first two calculations point out that here configuration interactions, coming from the one electron relativistic operators are insufficient to reproduce the experimen-
tal data and that core relaxation is negligible. In the
last row the total electrostatic interaction (direct and exchange parts) appears in the determination of the
wavefunction, leading to the value R30(o) = 3.26,
which is about midway between R th and the observed value R exp. The present result is very similar to the
one obtained in the study of the inverted fine-struc- tures [15]; indeed it was shown that the inversions
can be explained as being due to a second order cross
interaction between the spin-orbit perturbation and
the exchange part of the Coulomb interaction [20]
and that this approach is equivalent to a first order calculation of the central field approximation [16].
Moreover it is necessary to introduce the first 3 terms in the evaluation of Rn, indeed the discrepancy bet- ween R th and Rn(0) is of the same order of magnitude
as the difference between Rn(0) and Rn(2). This result
points out that the one- and two-electron excitations of the core electrons, which appear in the core relaxa-
tion, play a significant role. The values obtained for
Rn(2) are reported in column 3 of table 1 ; only 4 n
values (5, 10, 30 and 50) are evaluated, because the
corresponding calculations take a rather long time.
Finally, the calculated values obtained for RS(2) and R30(2) are in good agreement with the experimental
data [1, 2] and the ratio remains nearly constant for high n values. Contrary to the semi-empirical and
variational calculations, in which explicit correlation effects are negligible since they are introduced in an
effective way, in the present approach correlation and relativistic effects contribute simultaneously to the interpretation of the experimental data.
3. Conclusion. - Three different approaches from
the relativistic parametric-potential method have been
used to evaluate the oscillator strengths for the reso-
nance lines of rubidium. The semi-empirical calcu-
lation shows that the increase of the intensity ratio, Rn, with the principal quantum number, n, results from configuration interactions through the Breit-
1063
Pauli operators, which are automatically accounted
for in the first order calculation of the relativistic central field ; considering the simplicity of the model, qualitative agreement between calculated and observed
intensity ratios is reasonably good. The theoretical results are considerably improved in the variational and frozen-core approximations. In the latter treat-
ments the intensity ratios, and, furthermore the abso- lute f values, are in good agreement : the relative
discrepancy between the different numerical values obtained for a given f-value is not greater than 1.5,
even for the transitions to high Rydberg states where large cancellation effects occur in the calculation.
The comparison between the variational and the frozen-core approaches points out that relativistic and correlation effects are strongly connected and that the
separation usually made between these effects is not
unique and is somewhat arbitrary.
The present results show that, as n increases, the intensity ratio deviates rapidly from the ratio of the
statistical weights of the excited n 2Pj levels and reaches a limiting value, as can be predicted from the
classical theory of Fermi [3, 4]. For the limiting value,
the theoretical and experimental results are in good
agreement ; the result could be improved in the
framework of the frozen-core approximation by intro- ducing explicitly the modification of the core wave-
functions by the np electron into the optimization procedure giving the optimal central potential for the
core wavefunctions.
The present method could be used to study the
oscillator strengths for the principal series of caesium
and would probably lead to a limit value for the
intensity ratio for a sufficiently high value of n.
Acknowledgments. - The authors express their
acknowledgments to M.J.B. Johannin and his co-
workers for the exceptional facilities given in the computer centre Paris-Sud Informatique.
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