Asymptotic oscillator strength at the critical charge

Chemical Physics Letters 738 (2020) 136897

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Research paper

Asymptotic oscillator strength at the critical charge

T

Jacob Katriel Department of Chemistry, Technion – Israel Institute of Technology, Haifa 32000, Israel

HIGHLIGHTS

oscillator strengths are studied along isoelectronic sequences. • Atomic values at the critical charge are established. • Asymptotic transitions of an outermost electron these asymptotic values are hydrogenic. • For asymptotic quantum numbers depend on the occupancies of the inner shells. • The • Crude estimates of the oscillator strengths of realistic atoms are now feasible. ABSTRACT

A conjecture concerning the asymptotic behavior of the outermost orbital in atomic configurations with a single electron outside an N-electron closed shell, in the N , was proposed on the basis of an investigation of the asymptotic behavior of the corresponding quantum defect: If the outermost atomic orbital is of type limit Z n , then in the limit specified above it approaches an (infinitely diffuse) hydrogenic orbital of the form (n n ) , where n is the number of occupied subshells with angular momentum quantum number . The consequences of this conjecture concerning the corresponding asymptotic behavior of dipole and quadrupole oscillator strengths are formulated and explored. Reasonably convincing agreement is established for several transitions in the isoelectronic sequences of He, the alkali atoms, B, Al and Cu. Some apparent violations of the presently proposed asymptotic values of the oscillator strengths are pointed out.

1. Introduction Second only to the energies, transition probabilities among atomic quantum states have been of interest for about a century. They are considerably more difficult to determine accurately, both experimentally and theoretically. Atomic spectroscopy was at the forefront of the early development of quantum theory. Later interest has been primarily motivated by the needs of observational astrophysics and of plasma diagnostics. Highly accurate computations and measurements have in recent years been undertaken towards establishing properties that are relevant to the investigation of subtleties of nuclear structure as well as aspects of the Standard Model, such as parity violation [1]. The flavor of the present investigation may be reminiscent of early atomic theory. However, what justifies it is the rigorous understanding of the behavior of atomic ions upon approaching the critical charge, below which the outermost electron is not bound. To avoid any misunderstanding it is emphasized that the asymptotic systems implied are not experimentally realizable. They can only be accessed as extrapolations of systems with higher nuclear charges. Computationally, the critical charge can be approached as closely as desired, at an increasing cost. Experimentally, one can only extrapolate crudely, from measured values of oscillator strengths at physically allowed (integral) nuclear

charges. It is the latter approach which is presently applied. Let the number of electrons be N + 1. Two distinct scenarios have been established: The generic “expanding” scenario, in which the outN, ermost (singly occupied) orbital becomes infinitely diffuse as Z and the non-generic “absorbing” scenario, which is applicable to bound states of negative atomic ions. In this scenario the system remains Zc < N , where the binding energy of the square-integrable even as Z outermost electron vanishes [2]. In the present paper we are strictly concerned with the “expanding” scenario. For a single electron outside an atomic closed shell, the electric dipole oscillator strength for the transition from the lower level n to | = 1, is defined (in atomic units) as the higher level n , where | [3]

fn

,n

=

2 max( , ) (En 3 2 +1

En )

(

0

)

2

Rn rRn dr .

The radial functions Rn (r ) are defined according to the CondonShortley convention, satisfying 0 Rn2 dr = 1 (and similarly for Rn ). In a hydrogen-like atomic ion the energies (hence the energy differences) are quadratic in the nuclear charge Z, while the dipole matrix-elements Rn rRn dr are inversely proportional to Z. Hence, the oscillator 0 strengths are independent of Z.

E-mail address: [email protected]. https://doi.org/10.1016/j.cplett.2019.136897 Received 17 August 2019; Received in revised form 20 October 2019; Accepted 22 October 2019 Available online 26 October 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.

Chemical Physics Letters 738 (2020) 136897

J. Katriel

Several authors introduce the weighted oscillator strength gfn , n = (2 + 1) fn , n which is symmetric in the upper and lower states [4]. In some instances discussed below this distinction between the upper and the lower state will have to be dealt with explicitly. Within the isoelectronic sequence of a many-electron atom with a single electron outside a closed shell both the energies and the transition dipoles depend on the nuclear charge in a more complicated manner than in a hydrogen-like atomic ion. Using the framework of 1 -perturbation theory the oscillator strength can be written in the form Z [5]

fn

,n

= f n(0) , n +

1 (1) 1 f + 2 f n(2) , n + Z n ,n Z

Although the amount of oscillator strength data, both experimental and theoretical, is rather overwhelming, careful scrutiny suggests that there is ample room (and need) for further study that would provide more complete and in particular more accurate data. Convenient tabulations of hydrogenic transition matrix elements and oscillator strengths, that are needed in our analysis, are available in Green et al. [3], Herdan and Hughes [19], and Wiese et al. [20]. Their relativistic counterparts (whose values are very close to the nonrelativistic values for low Z) are presented by Pal’chikov [21] and by Jitrik and Bunge [22].

.

2. The asymptotic behavior of the oscillator strength

f n(0) , n is the hydrogenic oscillator strength and the following terms can be expressed within perturbation theory. Many atomic isoelectronic sequences have been investigated, both experimentally and theoretically. The high Z asymptotics were amply verified (although at sufficiently high Z relativistic effects play an increasingly significant role), e.g., [6,7]. Complications in the behavior of oscillator strengths along isoelectronic sequences have been reported, arising out of the appearance of Cooper minima [8,9], avoided level crossings and further configuration interaction. Such complications are sometimes heralded by the Z dependence of corresponding quantum defects, such as the d-orbitals quantum defects in the K-isoelectronic sequence, as opposed to the smooth behavior of the s and p quantum 18 limit [10]. defects, which approach the expected Z Here, we consider the low Z end of the isoelectronic sequence. Our argument is based on the following observation, that will be stated as a Conjecture: Consider an isoelectronic sequence with a single n electron outside an N-electron closed shell. Let n denote the number of -type N the outermost electronic subshells within the closed shell. In the limit Z orbital approaches an infinitely diffuse hydrogenic (n n ) orbital. For evidence that supports this conjecture, motivated by investigation of the behavior of the quantum defects in the Z N limit, we refer to [11–14]. In retrospect, Ivanov et al. [15,16], dealing with singlyexcited two-electron atoms, can be read as anticipating this conjecture. The consequences of this conjecture concerning the electric dipole and electric quadrupole oscillator strengths are stated in Section 2. The quantum defect approximation has been widely used to estimate atomic oscillator strengths [17,18], so the further step taken here, invoking the behavior of the quantum defect of an outermost electron at the critical charge, falls within an established framework. In Section 3 we examine data involving electric dipole oscillator strengths. A similar investigation is presented in Section 4 regarding electric quadrupole oscillator strengths, that are not nearly as abundant as their electric dipole counterparts. While enough sets of data have been located that allow tracing the behavior of the oscillator strength along the whole isoelectronic sequence and confirming the expected limiting behavior both N and as Z as Z , in many instances a variety of complicating N limit to be ascertained, and matters arise that do not allow the Z that deserve further scrutiny. In such cases only a computational investigation for non-integral Z values, approaching N, could establish the limiting behavior claimed above. Several authors have examined the behavior of atomic oscillator strengths along isoelectronic sequences, starting at the neutral atom. The high Z hydrogenic limit (ignoring relativistic effects) was invariably confirmed. When n = n the high Z oscillator strength vanishes. In such cases one can examine the high Z behavior of the squared dipole (or quadrupole) transition integral

(

0

Rn r Rn dr

)

2

=

fn 3(2 + 1) 2max( , ) En

,n

En

.

In the present section we state the implications of the behavior of N limit concerning the behavior of the the outermost orbital at the Z electric dipole and the electric quadrupole oscillator strengths at this limit. These statements are confirmed in the following two sections by examination of pertinent experimental and computed oscillator strengths. Typically, plots of the oscillator strength, appropriately scaled with respect to Z N , vs. 1 were examined. The limiting value as

Z

0 is easily verified rather precisely, because this point is an

1 Z

accumulation point of the sequence 1 . The limiting behavior as Z 1

Z

1

N

(or Z ) can only be estimated more crudely, so our claim that N verification was established just means a crude consistency of the curves with the predicted asymptotic values. A review of the cases examined is provided in Sections 3 and 4 for electric dipole and electric quadrupole oscillator strengths, respectively. In the interest of brevity and in view of the immediacy of the analysis involved only three illustrative figures are presented; For each of the other cases considered we provide a brief description of the pertinent features of the set of isoelectronic data examined. The sources of the data used are carefully quoted so that any particular case can fairly straightforwardly be verified. 2.1. Electric dipole transitions Applying the conjecture stated above to the evaluation of the oscillator strength, in the Z N limit, for an isoelectronic sequence with a single electron outside an N-electron closed shell, the simplest case n where the N-electron closed involves transitions of the form n shell does not contain electrons of types or . In such cases both the 1 0 and the Z N limits yield the hydrogenic oscillator strength Z

f n(0) , n . In between these limiting values of Z the oscillator strength goes through (at least one) extremum. Simple examples are transitions between p-type and d-type states in the Li or B isoelectronic sequences, or transitions between d- and f-type states in Na, Al, K. More generally, if the N-electron core (which is common to the initial and the final states) contains n subshells with angular momentum and n subshells with angular momentum , then lim gi fn

Z

N

,n

= gi f ((0) n

n ) ,(n n )

.

n can be Here, the state n is lower than n , so gi = 2 + 1, but n higher than n n , in which case gi = 2 + 1. Otherwise gi = gi = 2 + 1. n = n n , e. g., in the f ((0) vanishes when n n n ) ,(n n ) (1s 2p)1,3P (1s3s )1,3S isoelectronic sequences of He [n 0 = 1 and n1 = 0 , so the Z = 1 asymptotic outer orbitals are (infinitely diffuse) 2p and 2s , respectively]. In such cases a more informative low Z test of the asymptotic behavior can be obtained by using the device recommended above, multiplying the expression on the right-hand side of Eq. (1) by (Z N ) 2 , and extrapolating to Z N to obtain the squared hydrogenic transition dipole expected; near this limit Z N is the effective charge seen by the outermost electron.

(1)

A plot of the expression on the right-hand side of Eq. (1) multiplied by Z2 should converge towards the hydrogenic expression 2 R . The relevance of this expression or a ( 0 n r Rn dr ) at Z slightly modified form of it at the low Z limit is discussed below. 2

Chemical Physics Letters 738 (2020) 136897

J. Katriel

2.2. Transition between a closed shell and a singly excited state Atomic (N + 1) -electron closed shell systems characteristically remain bound as singly-negative ions, possessing non-integral critical charges Zc < N . The most familiar example is the two-electron atom, whose ground state remains bound (square-integrable) even at the critical charge, Zc 0.911028…[23]. In singly-excited states of such N, atoms the outermost electron becomes infinitely diffuse as Z scaling as 1 , whereas the ground state is still localized. ConseZ N quently, in this limit the dipole transition integral vanishes though the energy difference between the excited and the ground state remains finite, yielding a vanishing oscillator strength as Z N . This behavior was confirmed by examination of the oscillator strengths for the tran(1snp)1P , n = 2, 3, 4, 5, 6 [24]. Further confirmation sitions (1s 2 )1S follows from an asymptotic expression for the oscillator strength for the same transition proposed by Khandelwal et al. [25,26], that involves an effective charge Zi = Z for the 1s orbital in the initial configuration, 5 where , and an effective charge Zf Z 1 (that vanishes as 16 Z 1) for the np orbital in the final configuration. The asymptotic 1 and apexpression proposed by these authors vanishes as Z Z proaches the hydrogenic f1(0) as [27]. s np Somewhat crudely computed oscillator strengths for the Si isoelectronic sequence [28,29] suggest that the oscillator strength may vanish Z 12 (3p2 )3P0 (3p3d )3P1 at for the transitions and (3p2 )3P0 (3p3d )3D1. The transitions into the higher d states seem to exhibit Cooper minima, making a sound extrapolation to Z = 12 im(3pns )3P1 transitions, possible. The results for the (3p2 )3P0 n = 4, 5, 6, 7 , appear to be consistent with vanishing oscillator 12. strengths at Z

Fig. 1. The He isoelectronic sequence: The oscillator strengths presented, , f (0) f2s,2p , f2s,3p and f3s,4p are consistent with the Z 1 limiting values f1(0) s,2p 1s,3p and

f 2(0) s,4p , respectively.

Z = 2 and Z = 1. This definitely suggests that a computational investigation for fractional Z values approaching Z = 1 is called for. Such 43F 0 an extremum is also suggested by the low Z data for the 33D oscillator strength [30]. 3.2. Li isoelectronic sequence Oscillator strengths in the Li isoelectronic sequence have been studied rather extensively. Comparison with the critical values presented by Martin and Wiese [31] suggests that the conjecture proposed above nd oscillator strengths, n = 3, 4, 5, 6, for 3p nd holds for the 2p nd (n = 5, 6), as well as for the 2p ns , (n = 4, 5, 6), for 4p (n = 4, 5, 6) oscillator strengths. Fig. 2 illustrates two of these transitions. Qu, Wang and Li [32] considered the oscillator strengths for the nd, n = 3, 4, , 9 for 3 Z 10 . One notes that the transitions 2p n = 3 sequence exhibits a minimum at Z = 4 , rising (towards the hydrogenic value) both at the high Z limit (Z ) and at the low Z limit 2 ). The n = 4 sequence seems to exhibit a minimum at Z = 4 and (Z a maximum at Z = 7 . The conjectured Z = 2 limit suggests that the computed Z = 3 oscillator strength may possibly be about 1% too high. The n = 5, 6, 7, 8, 9 sequences exhibit maxima at Z = 4 , appearing to be consistent with the hydrogenic limit at both the high and the low Z np oscillator strengths limits. The extrapolation of the 2s 2 suggests considerably lower values than the (n = 3, 4, 5, 6 ) to Z np hydrogenic values. This discrepency is presented conjectured 1s with the hope that a reader of this article will be able to sort it out.

2.3. Electric quadrupole transitions Again, we consider transitions that involve a single electron outside a closed shell. Transitions that are forbidden in the electric dipole ap| = 2 , are facilitated by the electric proximation, such as = or | quadrupole interaction, yielding the oscillator strength

fn

,n

=

2

30(2

(En + 1)

En

)3 (

)

2

0

Rn r 2 Rn dr ,

where is the fine-structure constant. Since, in the hydrogenic approximation, the squared quadrupole matrix element scales as 14 and Z

the energy difference cubed scales as Z 6 , the quadrupole oscillator strength scales as Z 2 . To inspect the Z N limit we consider the scaled quadrupole osf cillator strength n ,n 2 . The behavior both as Z and as Z N is (Z

3.3. Boron isoelectronic sequence

N)

expected to be analogous to that of the dipole oscillator strength.

A fairly extensive study was presented by Lavin and Martin [33]. 3s transition are The data they present in their Fig. 1 for the 2p

3. Examination of asymptotic dipole oscillator strengths 3.1. He isoelectronic sequence A fairly rich set of pertinent data was provided by Cann and Thakkar (1snp)1P oscillator strengths, for [24]. Plots of the (1s 2s )1S 1 limiting oscillator n = 2, 3, 4, 5, 6 , vs. Z, are consistent with the Z np oscilstrengths being equal to the corresponding hydrogenic 1s (1snp)1P oscillator lator strength. In the same limit, the (1s3s )1S np oscillator strengths, for n = 4, 5, 6 , approach the hydrogenic 2s (1snp)1P oscillator strengths, for n = 5, 6, apstrength, the (1s 4s )1S np oscillator strength, and the proach the hydrogenic 3s (1s5s )1S (1s6p)1P oscillator strength approaches the hydrogenic 4s 6p oscillator strength. Three of these transitions are illustrated in Fig. 1. On the other hand, ignoring relativistic effects, the (1sn s )1 S (1snp)1P oscillator strength approaches the hydrogenic ns np oscillator strength at the Z limit. The corresponding triplet oscillator strengths seem to go through an extremum between

Fig. 2. The Li isoelectronic sequence: The oscillator strengths f2p,3d and f3p,4d are f (0) consistent with the hydrogenic f 2(0) p,3d and 3p,4d , respectively, both as Z

as Z

3

.

2 and

Chemical Physics Letters 738 (2020) 136897

J. Katriel f1(0) s,2p

0.13873. Note that the p orbital consistent with limZ 4 f2p,3s = 3 becomes, upon approaching Z = 4 , the higher orbital, requiring the statistical weight to be properly taken care of. 3.4. Na isoelectronic sequence The frozen-core approximation was used by McEachran, Tull and Cohen [34] to evaluate oscillator strengths for Na, Mg+ and Al++. Their fnd, mf values for n < m are in very good agreement with the

Z

(0) n 10 asymptotic value, f nd , mf . Their fnp, md values for

m are in

fairly good agreement with the expected f ((0) n 1) p, md limiting value. The fns, mp values only agree with the corresponding asymptotic values,

Fig. 3. The Cu isoelectronic sequence: The oscillator strengths f4s,4p and f4p,4d

f (0) are consistent with the hydrogenic f1(0) s,2p and 2p,3d , respectively, as Z

f ((0) n 2) s,(m 1) p to roughly a factor of two. This is, quite likely, a consequence of assuming a frozen core in the computation. An extensive set of oscillator strengths for the Na isoelectronic sequence, at the Hartree-Fock level, was presented by Biemont [35]. The graphical presentation of the oscillator strengths vs. 1 allows an easy 1

1

3.7. Cs isoelectronic sequence In the Cs atom the outermost orbital is 6s, but as Z is raised the outermost orbital becomes 5d at La++ and 4f at Ce+3 [39]. This rough 54 limit by inspecting behavior could mean that establishing the Z experimental (or computed) data along the physical (integer Z) isoelectronic sequence could be tricky. Calculated oscillator strengths for transitions involving the outermost electron in the first five members of the Cs isoelectronic sequence (Cs, Ba+, La++, Ce+3 and Pr+4) were reported by Migdalek and Wyrozumska [40]. Since the core sublevel occupancies are ns = 5, np = 4 and nd = 2 , we expect

Z

examination of the agreement with the Z = 10 extrapolated value. Transitions for which convincing agreement is observed (indicating the asymptotic transitions in paretheses) are

3s 3p 4p 3d

4p (1s 3d (2p 4d (3p 6f (3d

3p) 3d ) 4d ) 6f )

4s 3p 4p 4d

5p (2s 4d (2p 5d (3p 6f (4d

4p) 4d ) 5d ) 6f )

3p 3p 5p 5d

5s (2p 5d (2p 5d (4p 6f (5d

28.

3s ) 5d ) 5d ) 6f )

An interesting comparison, establishing that oscillator strengths of n s); K homologous transitions in alkali atoms [i. e., Na (np ((n + 1) p (n + 1) s ) ; ((n + 2) p (n + 2) s ) ; Rb Cs ((n + 3) p (n + 3) s ) ] are rather similar, was presented by Gruzdev [36]. Gruzdev claimed that for several such transitions the oscillator n s ) transition. The strengths are close to those of the hydrogenic (np asymptotic behavior suggested in the present paper would prefer a (n 2) s ) transition [becomparison to the hydrogenic ((n 1) p cause there is one inner p electron and two inner s electrons in the Na core; the number of inner p and s electrons in the homologous alkalis consecutively growing by unity]. Indeed, in most cases this modified comparison yields better agreement with the corresponding alkali oscillator strengths.

lim f (6s

6p)= f (0) (1s

2p)

0.416198

lim f (6p

5d )= f (0)

(2p

3d )

0.695784

lim f (6p

6d )= f (0) (2p

4d )

0.121795

lim f (6p

7d )= f (0) (2p

5d )

0.044370

lim f (5d

4f )= f (0) (3d

4f )

1.0175

lim f (5d

5f )= f (0)

(3d

5f )

0.15664

lim f (5d

6f )= f (0)

(3d

6f )

0.05389.

Z Z Z Z

Z Z Z

54

54 54 54

54 54 54

Agreement with the values reported in [40], in most cases to within 10%, is observed. 54 , where 4f (0) as Z 1s (0), 5d 3d (0) and 4f The fact that 6s the zero superscript emphasizes the hydrogenic character of the limiting orbital, i. e., 6s < 5d < 4f , can be taken to account for the orbital ordering in the neutral atom. The 6s 5d and eventually the 5d 4f inversions simply reflect the approach towards hydrogenic ordering at the high Z limit.

3.5. Al isoelectronic sequence

3d was studied (among many The transition (1s 2 2s 2 2p6 3s 2 );3p other transitions, that do not suit our one-electron framework) by Safronova et al. [37]. The oscillator strengths are reported starting at Z = 15 (The neutral Al atom has 13 electrons). The expected asymptotic 0.695784 , is (somewhat oscillator strength, limZ 12 f3p,3d = f 2(0) p,3d roughly) consistent with the reported values. As Z increases f3p,3d decreases, appearing to proceed towards zero at large Z. However, relativistic effects eventually take over and the oscillator strength starts increasing.

4. Quadrupole oscillator strengths: the He isoelectronic sequence The set of computed quadrupole oscillator strengths provided by Cann and Thakkar for the He isoelectronic sequence [41] allows an 1 limit for the n1,3S m1,3D and for the examination of the Z n1,3P m1,3P transitions (where n < m ). Since the core consists of a (single) 1s electron, the quadrupole moments are expected to satisfy

3.6. Cu isoelectronic sequence

lim

Z

Semiempirically computed oscillator strengths for the Cu isoelectronic sequence, whose core is specified by ns = 3, np = 2, nd = 1, were 4p (1s 2p), presented by Victor and Taylor [38]. The 4s 4p 4d (2p 3d ), 4p 6s (2p 3s ), 4p 7s (2p 4s ) transitions 28 limits. The same is true for yield convincing Z 4f (3d 4f ), 4d 5f (3d 5f ), 5d 5f (4d 5f ) , 4d 6f the 4d 6f (4d 6f ) oscillator strengths. Two of these (3d 6f ) and 5d transitions are illustrated in Fig. 3.

1

f (n1,3S (Z

m1,3D ) = f (0) ((n 1) 2

f (n1,3P (Z

m1,3P ) = f (0) (n1,3P 1) 2

1)1,3S

m1,3D)

and

lim

Z

1

m1,3P ),

while

lim

Z

4

f (n1,3S

m1,3D) Z2

= f (0) (n1,3S

m1,3D)

Chemical Physics Letters 738 (2020) 136897

J. Katriel

and

lim

f

(n1,3P

Z

m1,3P ) Z2

= f (0) (n1,3P

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m1,3P ).

The latter two relations were confirmed by Cann and Thakkar [41]. The 1 extrapolation required to establish the limiting behavior at Z cannot be carried out with a high degree of precision, but consistency is suggested by the curves of the scaled quadrupole oscillator strengths plotted against 1 (or against Z). The n1,3P (n + 1)1,3P (n = 2, 3, 4, 5) Z curves exhibit slight convexity (obtaining maxima), the triplet oscillator (n + 2)1,3P (n = 3, 4 ) strength being above the singlet. The n1,3P curves behave in a very similar way. However, for n = 2 the triplet quadrupole oscillator strength is lower than the corresponding singlet quadrupole oscillator strength for Z < 6 , possibly going through an 51,3P , the inflection point at Z 1. The same holds for the 21,3P 1,3 1,3 1,3 1,3 2 P 6 P , and the 3 P 6 P quadrupole oscillator strengths. In these cases it appears that the singlet-triplet average tends more con1 1 limit (where the singlet and the triplet vincingly towards the Z coincide, just as in the Z limit). 5. Conclusion For electric dipole or electric quadrupole transitions in which a single outermost electron outside an N-electron core is excited from an n into an n orbital, it was argued that in the limit Z N the oscillator strength approaches a hydrogenic oscillator strength between n ) and an (n n ) orbital. Here, n and n are the number an (n of subshells with angular momentum and , respectively, in the core. n , if > it is possible that n > n , hence While we assume that n n n n . In such cases one should be careful to compare that n weighted oscillator strengths, that are symmetrical between the lower and the higher state. A reasonable set of isoelectronic data was examined to substantiate this claim. Declaration of Competing Interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] A.A. Vasilyev, I.M. Savukov, M.S. Safronova, H.G. Berry, Measurement of the 6s7p transition probabilities in atomic cesium and a revised value for the weak charge Q_W, Phys. Rev. A 66 (2002) 020101. [2] J. Bellazzini, R.L. Frank, E.H. Lieb, R. Seiringer, Existence of ground states for negative ions at the binding threshold, Rev. Math. Phys. 26 (2014) 1350021. [3] L.C. Green, P.P. Rush, C.D. Chandler, Oscillator strengths and matrix elements for the electric dipole moment for hydrogen, Astrophys. J. Supp. 3 (1957) 37. [4] A. Farrag, E. Luc-Koenig, J. Sinzelle, Relativistic oscillator strengths in the boron isoelectronic sequence, At. Data Nucl. Data Tabl. 24 (1979) 227. [5] A. Dalgarno, The Z-independence of oscillator strengths, Nucl. Instr. Meth. 110 (1973) 183. [6] W.L. Wiese, A.W. Weiss, Regularities in atomic oscillator strengths, Phys. Rev. 175 (1968) 50. [7] M.W. Smith, G.A. Martin, W.L. Wiese, Systematic trends and atomic oscillator strengths, Nucl. Instr. Meth. 110 (1973) 219. [8] U. Fano, J.W. Cooper, Spectral distribution of atomic oscillator strengths, Rev. Mod. Phys. 40 (1968) 441. [9] C.E. Theodosiou, Minima in the emission oscillator strengths of alkali Rydberg

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