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 rubidium : relativistic and correlation effects. Journal de Physique, 1978, 39 (10), pp.1059-1063.
<10.1051/jphys:0197800390100105900>. <jpa-00208845>
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LE JOURNAL DE
PHYSIQUE
TOME
39,
OCTOBRE
1978,
1059
Classification
Physics Abstracts
32.70
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.
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 relativistes 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.
2014
=
=
Abstract.
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 correlation 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
on highly excited Rydberg
2014
=
=
experiment
states.
1. Introduction.
Recently anomalous intensities Rn increases with n and tends towards a limit value [4],
have been observed in the fine-structure patterns for as the admixture with the lowest n ’Pj levels contrithe 5s-np transitions (29 n 50) of Rbl [1]. For butes the predominant part of the departure of Rn
from R th. Further theoretical or experimental invesall patterns which have been measured the ratio Rn
of the intensities of the two components In3/2 and tigations of the ratio for high lying levels lead to diffeInl/2 of the doublet differs from the ratio of the statis- rent predictions for the behaviour of Rn : from some
results Rn would pass through a maximum as n
tical weights of the n
and n 2P 1/2 excited levels :
increases
is
to
the
observed
value
instead of R th
[5-8], from the others the ratio would keep
2,
equal
an increasing [9-12]. No experimental data are avaiRexp = 5.9 + 1.4 and, within the experimental uncertainty, does not depend on n. However no experimen- lable for n &#x3E; 19, and in the most recent experiments
tal data are available for the low lying levels, except [9, 10, 13] no maximum was found for the ratio.
5 where R5exp
1.99 ± 0.34 [2]. Therefore Several theoretical calculations of the oscillator
for n
there is no experimental information on the evolution strengths for the principal series of alkali-metals
have been performed using non relativistic wave6 and n
of Rn between n
28.
Similar anomalies were observed for the first time functions obtained from semi-empirical model potenin 1930 for the resonance lines of caesium. They were tials introducing spin-orbit and core polarization
explained qualitatively by Fermi [3] as resulting from effects [8, 12, 14]. The matrix elements associated
the strong non-diagonal spin-orbit interaction bet- with transitions to highly excited levels are very
sensitive to the part of the wavefunctions near the
ween the n 2Pj levels. To first order in perturbation
the
contributions
to
and
f3/2
theory,
fn1/2 coming nucleus, where the overlap of the 5s and np orbitals is
from the excited levels m, such as Em &#x3E; En, tend to not negligible. Consequently Norcross’s calculations
reduce Rn, but these contributions decrease rapidly [12] give the more reliable results for CsI, since they
with m ; on the other hand the perturbing levels m introduce, besides the spin-orbit, additional relatiwith Em En correspond to a positive correction to vistic corrections which are associated with the proper
Rn. The perturbations coming from the low lying behaviour of the relativistic potential in the vicinity
levels are predominant, since these levels are associated of the nucleus.
The oscillator strengths for the principal series of
with the strongest transition matrix elements. Thus
-
2P3J2
=
=
=
=
=
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 polarization 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=
=
=
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
transition 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 expression -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 approximation. 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
=
2S1/2-n 2P 1/2
TABLE 1
Intensity ratios for the
resonance
lines
of rubidium
order calculation of configuration interactions through
the one-electron Pauli operators (Darwin, p4 term,
spin-orbit) [17]. The anomalous Landé factors observed experimentally in the sp configurations of Cd and
Hg can likewise be reproduced in a first order calculation of the relativistic central field approximation
[18] ; in that example the relativistic corrections to the
operator obtained through the use of the FoldyWouthuysen transformation are almost negligible,
while the relativistic corrections to the wavefunctions 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 introduced in a first order calculation of the relativisticcentral-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 separation is somewhat arbitrary.
Intensity ratios for the resonance lines of rubidium.
w) Weisheit’calculation, Ref. [8].
a) semi-empirical calculation
b) variational calculation
this work.
c) frozen-core calculation
Experiment : hatched area R exp 5.9 ± 1.4, Ref. [1].
FIG. 1.
-
Theory :
1
=
2. Relativistic calculations of the oscillator strengths
The relativistic
for the principal séries of rubidium.
radial wavefunctions are obtained from the relativisticparametric-potential method [15] and the velocity formulation for the oscillator strengths is used.
-
1061
We have used the
velocity
form of the transition
operator since this operator samples the wavefunctions
relatively small r values. Indeed the different approaches 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 wavefunction is negligible beyond the 12th node of the
excited np wavefunction (n &#x3E; 15) so that the transition matrix elements do not allow to test the excited
wavefunctions at large r values.
at
2.1 SEMI-EMPIRICAL CALCULATION FROM A MODELThe central potential is represented by
an analytic function which depends on three parameters. The long-range core polarization effects are
introduced in an effective way by adding to the central
potential the correction
POTENTIAL.
-
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 corresponding 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 introduces in an effective way and on an average all configuration 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 experimental 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.
T 0 take into
2.2 VARIATIONAL CALCULATION.
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
level. We have
chosen the 10
level which is sufficiently excited
to represent the effect of high lying Rydberg p states ;
-
2p 3/2
2Pj
’P3/2
for the 10
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
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
moreover
(-
2p 1/2
integrals (nljn’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 nondiagonal integrals represent approximately the contribution 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 relative 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
resonance
of the intensity ratios
lines of rubidium
for
the
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. Moreover the limiting value 4.7 is greater than in the preceding approach and does not vary significantly for
20 n 80. Experimental and theoretical results
are in good agreement pointing out that core relaxation effects cannot be neglected.
TABLE III
=
=
Frozen-core
approximation
=
2. 3 FROZEN - CORE APPROXIMATION.
In this
the
5s
state
is kept
wavefunction
approach
ground
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 wavefunctions 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 conditions between core and valence electrons were automatically 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 procedure 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 approximations 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 insufficient 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-
=
=
=
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 experimental 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-structures [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 between 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 relaxation, 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.
=
Three different approaches from
3. Conclusion.
the relativistic parametric-potential method have been
used to evaluate the oscillator strengths for the resonance lines of rubidium. The semi-empirical calculation 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 treatments the intensity ratios, and, furthermore the absolute 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 introducing explicitly the modification of the core wavefunctions 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 coworkers for the exceptional facilities given in the
computer centre Paris-Sud Informatique.
-
References
[1]
[2]
[3]
[4]
[5]
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