- Email: [email protected]
Relativistic effects in transitions of highly ionized heavy atoms
Volume 20, number 5
CHEhfIC4L
1 July 1973
PHYSICS LETTERS
RELATIVISTIC EFFECTS IN TRANSITIONS OF HIGHLY IONIZED HEAVY ATOMS Oktay SINANOCLU Sterling Chemistry Laboratory.
and William LUKEN
Yale University, New lfawtz, Connecticut
06520, USA
Received 30 April 1973
Wavelengths, oscillator strengths, and lifetimes of KL shell heavy ions such as FeZW, Br30c, etc. arc predicted, including both relativistic and corrclstion effects. In such ions the distinc:ion between “normal” electric dipoIe (El) and spin-orbit allowed electric dipole transitions (SOAEI) disappears and the Zdependence of oscillator strengths is very different from the predictions of non-relativistic orbital theories. The feasibtii?y of obreming these ions in heavy ion acceleratofi is indicated. The results have implications for relativistic many-electron theory, ESCA, X-ray spectra, and molecular total energies.
It has recently been shown that orbital or HartreeFock (RHF) methods give the wrong behaviour of isoelectronic electric dipole oscillator strengths.fEi, as the atomic number, Z, approaches the neutral atom [ 1) As shown in figs.. 1 and 2, the f-values frequently dip down sharply at the neutral end due to important open-sheil correlation effects first discovered with the non-closed shell many-electron theory (NCMET) of Sinano3u. At the high Z end of isoelectronic sequences however, it is generally thought that orbital theories, even hydrogenic ones [2], should work quite well, witlrf going to zero linearly with 1/Z for in-shell transitions. This expectation is based on the fact that correlation effects become smaller compared to the orbital effects as I/Z approaches zero. Such curves are therefore often used to extrapolate f-values from lower Z’s In the present letter, we show that for KL+ KL’ transitions, relativistic effects cause fversus 1 /Z curves to deviate strongly from the linear approximation for Z’s greater than 25> and in some cases for Z’s as low as Si. In these ions, the large spin-orbit interaction destroys the LS multiplet structure, requiring the use of intermediate coupling. The results indicate that such ions would be observable in heavy ion accelerators such as the SUPER-HILAC at Berkeley, coupled with the beam foil spectroscopy technique.
For open-shell many-electron ions, two types of relativistic effects are important: (i) orbital effects. and (ii) configurational effects. The first are mainly the result of the orbital shrinkage caused by the relativistic mass increase. This produces a similar reduction in the transition dipole, but for 2 < 50, this is a small effect which will be neglected in this approdmation [3]. This shrinkage also causes a change in the orbital energies. For transitions with a change in principal quantum number (e.g.. is+2p) this effect is small? however for in-shell transitions (e.g.. 3-c 2p), the electrostatic energies are nearly equal, and the relativistic contributions become very important. The relativistic configurationai effects are caused by the spin-orbit interaction. which becomes comparable to the electrostatic interaction in these ions. As a result, the intermediate coupling hamiltonian: H IC = HLS + %O
(1)
must be diagonahzed separately for each Z. This recoupling has dramatic effects on both line strengths and wavelengths. In addition, since S is no longer a good quantum number, the selection rule &S = 0 no longer holds, and transitions of the “spin-orbit allowed electric dipole” (SOAE1) type become formally identical to the “nor-mill” eiectric dipole (El) transitions. 407
Volume 20, number S
III 9) 45
1 July 1973
CHEMICAL PHYSICS LETTERS
I 15 112
III1 Ne 0 N
I C
0.0 f
B
*
+
,;-7 *-,= , 45
1
*
,
,
15 Nei3NC I/Z +
f
B
Fig. 1. Boron isoelectronic sequence oscillator strengths versus l/Z. The solid curve represents the 2s22p (J=1/2)-2~2~’ (J=3/2, P3) transition and the broken curve represents the 2s22p (J=1/2)+2s2pz (J=3/2, #2) transition. The low Z (Z < 10) end of the solid curve corresponds to the 2s22p 2 P(ljZ -+ 2s2p2 2 P3,2 transition, and the iow 2 end of the broken curve correspondsto the 2s22p 2Py1242s2p2 *D3,2 transition. The low Z values were obtained from non-relativistic calculations using the non-closed shell many-electron theory (NCMET) of Sinanoilu (ref. [ 1 ] ). The high Z values are from this work, with intermediate coupling upper states composed of 4Pz,2, ‘DJ,* and ‘Q2 terms.
Fig. 2. Boron isoelectronic sequence oscillator strengths versus l/Z. The solid curve represents the 2s22p (J=1/2-2s2p2 (J=1/2, +3) transition, and the broken curve represents the 2s22p (J=1/2)+2s2pz (J=1/2, R3) transition. The low Z (Z < 10) end of the solid curve corresponds to the 2s22p 2P0*,*+2s2p2 2P 1,2 transition, and the low Z end of the broken c3rvc corresponds to the 2s2 2p 2Py,z -2s2p2 ’ SI12 transition. The low Z values were obtained from non-relativistic calculations using the non-closed shell many-electron theory (NChlET) of SinanoHlu (ref. [ 1] ). The high Z values
The example of boron sequence at high Z 3: the above considerations have been applied as an example to transitions between the ls22s22p and 1s22s2p2 configurations in the boron sequence. Elements of H,, were obtained by fitting an equation of the form [4]
each value of Z to obtain intermediate coupling eigenvalues and eigenvectors. Each eigenvector contains a single LS term for the ls22s22p configuration and up to three LS terms in the 1~22~2~2 configuration. A few sample eigenvectors are shown in table 1. Differences in the eigenvalues were used to calculate wavelengths.. Non-empirical LS multiplet strengths for each of ‘LheLS allowed transit&s between the ls22s22p and ls22s2p?- configurations were calculated for silicon using the new atomic structure theory of Sinanogu [I]. These LSJ multiplet strengths, which include the important “non-dynamical” correlation effects of NCMET, were converted to LSJ line strengths and extrapolated to high Z’s with equations of the Form s (,Z) = z(zo) Z$Z” . intermediate coupling line strengths were then calculated from these LSJ line strengths and the eigenvectors of eq. (1) by use of the equation
E(LS,Z) X K*(L
= c, (L,S) f [Z-Z, S) + [Z -Zg(L
(Ls)] S)l C,(L,
(2) 91
to empirical low Z data [S] (,Z=S to Z=lS) and extrapolating to high 2’s. Elements of HSO were obtained using the matrices given in Condon and Shortley [6] and spin-orbit coupling parameters obtained by fitting an equation of the form [4] t(Z) = Cso (Z-
Zso)”
(3)
to empirical low Z rirre structure data [S]. The parameters obtained are Cso = 0.246 cm-l and Zso -- ’_.& 192 for the ls22s22p configuration and Cgo = 0.248 cm-l and Z&, = 2.479 for the ls*2s2p2 configuration. HI, was then obtained from eq. (1) and diagonalized at 408 ‘.
are from this York, with intermediate coupling upper states composed
of P1,2, 2s 1~2, and ‘PI,2 terms.
= ~~~C(L2,S~;J,)S1n(L,, t ,
S,.J, -+L,,S,,
J2)
2
.
Volume 20, number
5
CHEhlICAL PHYSICS LETTERS
Table 1 Intermediate coupling eigcnvalues and eigenvectors states of the ls22s2p2 configuration”). P z=15 E(31 2PI:2 %/2 4P1,2
E(2) ZP1/2 2s112
4PI/2 E(1) 2P1/2 2.5,,2
4P1/2
hln z=25
Br z=35
for J=l/2
Table 2 Transition energies, A E, line strengths, S, and spontaneous emission rates, A, in intermediate coupling for 2sz2po+ . 2s2p’ transitions a)
Rh Z=45
427000 0.9794 0.20 18 0.0113
889000 0.7459 0.6654 -0.0304
1711000 0.5758 0.8045 -0.1457
3257000 0.4538 0.8496 -0.2681
403000 -0.2013 0.9789 -0.0347
800000 -0.6568 0.7271 -0.1997
1415000 0.7684 -0.4717 0.4326
2538000 0.7984 -0.2538 0.5460
178000 -0.0180 0.0317 0.9993
366000 -0.1107 0.1689 0.9794
525000 -0.2793 0.3611 0.8897
4 32000 -0.3957 0.4623 0.7935
F Z=lS ‘P912 -+(3)1,2 AE 427000 0.197 s A 16 2EL2
AE s A
A
cigenvalues.
AE S
The Condon-Shortley phase convention was observed in taking the square roots. Once the wavelengths, X , and line slrengths, S, had been calculated, oscillator strengths, f. spontaneous transition probabilities, A, and lifetimes, 7, were calculated from them. Some AE’s, S’s and A’s are shown in table 2. These results do not in&de a!1 possible KL+KL’ transitions, but they represent the type and magnitude of the effects involved. Similar effects will also be found in KLM, etc., in-shell transitions. The f versus l/Z curves that result from these calculations are very different from earlier predictions [2]. In figs. I and 2, typical (non-relativistic) openshell correlation effects are seen at the low 2 ends of the curves. In fig. 1, as the 2Pj,2 and ‘D3/2 mix due to spin-orbit effects, one curve “steals intensity” from the other. An “avoided crossing” of the 2P1,2 and 2S,,? IeveIs causes the behaviour shown in tig. 2. Some of the curves in figs. 1 and 2 turn up sharply for 2 > 50 due to the rapidly increasing (2”) energy differences. Fig. 3 shows a transition corresponding to the SOAEl type. This curve goes to zero for low 2’s. As can be seen in table 1, LS terms mix so heavily for 2 > 30 that there is no longer any distinction between
+
&In Z=25
Br z=35
_Rh Z=45
889000
1711000 0.032 161
3257OOi) 0.016 5.50
1285000 0.0015 3.2
2OL9000 0.0022 18
1415000 0.079 23
‘538000 0.080 I32
526000 2.0 X f04 3 x 10-Z
4;zooo 3.6 x IO+ 3 x 10-Z
0.074 53
(3)1,2
417000 0.048
790000
4XlOd 0.002
3.5
2Pp,,(2)1!2 ‘ AE S
E(2), and E(3) represent the first, second, and third eigenvslues respectively, in units of cm-‘. The coefficients of each LS term in each eigenvector are given below the
‘) E(l),
I July 1973
403000 0.019 1.3
800000 0.039
2.0
2pP,2 +(1)1/z A
178000 4.3 x 10-S 2 x lo-’
366000 1.8 x 10-s 9 x Lo-4
a! Line strengths are in atomic units, A’s are in nsec-I_ Upper states nre numbered 1, 2, 3 in order of increasing energy for each value of the total angular momentum J.
0””
’
“I’
I
I
’
’
-:
T-* - -6 $ -8
i
-10
--__ _ III
m
45
15
l/Z
II
I
Hf3’ONc
I B
_3
Fig. 3. Boron isoelectronic sequence oscillator strengths for spin-orbit allowed electric dipolc (SOAEI) transitions versus 1/Z. The solid curve represents the 2s22p (J=3/2)+2s2pz (&l/2, +l) transition, and the broken curve represents the 2s22p (J=1/2)+2s2p2 (5=3/2, +1) transition. The low Z (Z 6 10) end of the solid curve corresponds ta the 2s22p ZP9,2 * 2s2p2 4P,,* transition, and the low 2 end of the broken curve corresponds to the 2s22p ‘PO3,2+2s2p2 4P3;2 Lransition. All values obtained from this work using intermediate coupling upper states containing 4P. 2D. 2P: and ‘1~’ terms.
409
Vo!umc 20, numbx
5
CHEMICAL PHYSICS LETTERS
El and SOAEi iransitions. In the method used here both types are treated as one and both may have comparable values. In addition to the spin-orbi: interaction there are other related effects such as the “spin-spin” and “spin-other-orbit” interactions [7,8]. However, these effects are small compared to the spin-orbit in neutral atoms, and they grow only as Z3 rather than Z4. Consequently they become less important in high 2 ions and they can be neglected in the current work. Even so, these effecrs only represent an expansion to first order of a Dirac-like hamiltonian in terms of c?Z2. Beyond Z=60, an expansion in a2Z2 can no longer be expected to be rapidly convergent, as assumed here. In addition, quantum field theoretic effects (“radiative corrections”) which come in as (QZ)~. finite nucleus effects, hyperfine interactions, etc., may become important. Because a relativistic N-electron hamiltonian which includes all of these effects has not yet been developed, it is impossibie to carry these calculations beyond Z z 60. Unlike in the non-relativistic case, it is currently impossible to predict the asymptotic behaviour of these calculations as Z approaches infinity. The accuracy of these calculations is expected to range from about 10% for Z= 15 to factor of two or SO at 2~50, with certain exceptions. The curves in figs. I and 2 are drawn out to Z=65 to illustrate the behaviour of this model, but these last points are likely td he seriously in error. High-Z-low-N ions such as the ones discussed here can be produced in the laboratory by passing a less highly ionized beam of atoms through a thin carbon foil. The beam velocity required can be calculated from the semi-empirical formula of Dmitriev and Nicolaev [9, lo]. The corresponding beam energies arc ob!ainable for many atoms on Tandem Van de Graaff accelerators and the “SUPER-HILAC” at Berkeley. The transitions of these ions can then be studied by the technique of beam-foil spectroscopy. provided the distances travelled by the excited ions fall within measurable limits, typically on the order of one meter > VT > one millimeter [ 1 I]. The velocities calculated from ref. [lo], and transition rates obtained using the method described above predict that inany tIansitions decay within the measurable limits. .indicating the feasibility of observing these ions in the laboratory. ’ Any knowledge that wou!d be gained by the study 410...
.. .. ..: :
,’
1 July 1973
of these high-Z-low-N ions would be useful in developing a relativistic many-electron theory. Such a theory would be useful in chemistry because (a) due to the orbital orthogonality, relativistic effects of inner shell electrons are passed on to outer electrons, makmgi such effects important in less highly ionized and neutral atoms. (b) Inatomic and molecular calculations where the variational principle is applied, one needs experimental total energies for comparison with calculations. These experimental energies, which are needed also in obtaining the “experimental correlation energies”, involve a sum over all the ionization potentials of the atom(s) involved, and for heavy atoms most of these have not been observed. The theory and experiments discussed here would help fdl in the gaps in such data. (c) In addition, Auger and CosterKronig pro:esses. ESCA, and X-ray photoelectron spectroscopy frequently involve trarlsitions in which the initial znd/or final states have open L-shells similar to the ones considered above. The properties such as transition probabilities, inner binding energies, ESCA. splittings, etc., will all be affected by the same effects . presented here. This work was supported by a grant for the U.S. National Science Foundation. We thank Professor J. Rasmussen and Dr. Donald R. Beck for helpful dis-
cilssions. References Ill 0. Sinano$lu, Atomic physics, Vol. 1 (Plenum Press, New York, 1969) p. 131; P. Westhaus and 0. SinanoHlu, Phys. Rev. 183 (1969) 45; 0. Sinanogu, NucI. Instr. .?lcthods, 3rd Beam Foil SpectrOsCOpy COnfCrCnCe ISSUC(1973). I21 W.L. Wiese and .4.W. \Veiss, Phys. Rev. 175 (1968) 50. 131 R.H. Gxstang, in: Topics in modern physics, eds. WJZ. Btittin and H. Odabasi (Colorado Assoc. Univ. Press, Boulder, 197 1). [41 B. Edle’n, in: Handbuch der Physik, Vol. 27, cd. S. F16ge (Springer, Berlin, 1964) p. 156. [51 C. hloore, Atomic energy 1~~~1s.Vol. 1, NBS Circular 467 (US Govt. Ptinting Office, Washington, 1949). [61 E.U. Condon and G.H. Shortley, The theory of atomic spectra (Cambridge Univ. Press, London, 1935). I71 M. Blunie and R.E. Watson, Proc. Roy. Sot. A270 (1962) 127; A271 (1963) 565. ial R.E. Trxs, Phys. Rev. 82 (1951) 683. 191 H.D. Betz, Rev. Mod. Phys. 44 (1972) 465. 1101 I.S. Dmitriev and V.S. Nikolaev, Phys. Letters 28A (1968) 277. 1111 R. Marrus and R_ Schmiider, Phys. Rev. A5 (1972) 1160.
CHEhfIC4L
1 July 1973
PHYSICS LETTERS
RELATIVISTIC EFFECTS IN TRANSITIONS OF HIGHLY IONIZED HEAVY ATOMS Oktay SINANOCLU Sterling Chemistry Laboratory.
and William LUKEN
Yale University, New lfawtz, Connecticut
06520, USA
Received 30 April 1973
Wavelengths, oscillator strengths, and lifetimes of KL shell heavy ions such as FeZW, Br30c, etc. arc predicted, including both relativistic and corrclstion effects. In such ions the distinc:ion between “normal” electric dipoIe (El) and spin-orbit allowed electric dipole transitions (SOAEI) disappears and the Zdependence of oscillator strengths is very different from the predictions of non-relativistic orbital theories. The feasibtii?y of obreming these ions in heavy ion acceleratofi is indicated. The results have implications for relativistic many-electron theory, ESCA, X-ray spectra, and molecular total energies.
It has recently been shown that orbital or HartreeFock (RHF) methods give the wrong behaviour of isoelectronic electric dipole oscillator strengths.fEi, as the atomic number, Z, approaches the neutral atom [ 1) As shown in figs.. 1 and 2, the f-values frequently dip down sharply at the neutral end due to important open-sheil correlation effects first discovered with the non-closed shell many-electron theory (NCMET) of Sinano3u. At the high Z end of isoelectronic sequences however, it is generally thought that orbital theories, even hydrogenic ones [2], should work quite well, witlrf going to zero linearly with 1/Z for in-shell transitions. This expectation is based on the fact that correlation effects become smaller compared to the orbital effects as I/Z approaches zero. Such curves are therefore often used to extrapolate f-values from lower Z’s In the present letter, we show that for KL+ KL’ transitions, relativistic effects cause fversus 1 /Z curves to deviate strongly from the linear approximation for Z’s greater than 25> and in some cases for Z’s as low as Si. In these ions, the large spin-orbit interaction destroys the LS multiplet structure, requiring the use of intermediate coupling. The results indicate that such ions would be observable in heavy ion accelerators such as the SUPER-HILAC at Berkeley, coupled with the beam foil spectroscopy technique.
For open-shell many-electron ions, two types of relativistic effects are important: (i) orbital effects. and (ii) configurational effects. The first are mainly the result of the orbital shrinkage caused by the relativistic mass increase. This produces a similar reduction in the transition dipole, but for 2 < 50, this is a small effect which will be neglected in this approdmation [3]. This shrinkage also causes a change in the orbital energies. For transitions with a change in principal quantum number (e.g.. is+2p) this effect is small? however for in-shell transitions (e.g.. 3-c 2p), the electrostatic energies are nearly equal, and the relativistic contributions become very important. The relativistic configurationai effects are caused by the spin-orbit interaction. which becomes comparable to the electrostatic interaction in these ions. As a result, the intermediate coupling hamiltonian: H IC = HLS + %O
(1)
must be diagonahzed separately for each Z. This recoupling has dramatic effects on both line strengths and wavelengths. In addition, since S is no longer a good quantum number, the selection rule &S = 0 no longer holds, and transitions of the “spin-orbit allowed electric dipole” (SOAE1) type become formally identical to the “nor-mill” eiectric dipole (El) transitions. 407
Volume 20, number S
III 9) 45
1 July 1973
CHEMICAL PHYSICS LETTERS
I 15 112
III1 Ne 0 N
I C
0.0 f
B
*
+
,;-7 *-,= , 45
1
*
,
,
15 Nei3NC I/Z +
f
B
Fig. 1. Boron isoelectronic sequence oscillator strengths versus l/Z. The solid curve represents the 2s22p (J=1/2)-2~2~’ (J=3/2, P3) transition and the broken curve represents the 2s22p (J=1/2)+2s2pz (J=3/2, #2) transition. The low Z (Z < 10) end of the solid curve corresponds to the 2s22p 2 P(ljZ -+ 2s2p2 2 P3,2 transition, and the iow 2 end of the broken curve correspondsto the 2s22p 2Py1242s2p2 *D3,2 transition. The low Z values were obtained from non-relativistic calculations using the non-closed shell many-electron theory (NCMET) of Sinanoilu (ref. [ 1 ] ). The high Z values are from this work, with intermediate coupling upper states composed of 4Pz,2, ‘DJ,* and ‘Q2 terms.
Fig. 2. Boron isoelectronic sequence oscillator strengths versus l/Z. The solid curve represents the 2s22p (J=1/2-2s2p2 (J=1/2, +3) transition, and the broken curve represents the 2s22p (J=1/2)+2s2pz (J=1/2, R3) transition. The low Z (Z < 10) end of the solid curve corresponds to the 2s22p 2P0*,*+2s2p2 2P 1,2 transition, and the low Z end of the broken c3rvc corresponds to the 2s2 2p 2Py,z -2s2p2 ’ SI12 transition. The low Z values were obtained from non-relativistic calculations using the non-closed shell many-electron theory (NChlET) of SinanoHlu (ref. [ 1] ). The high Z values
The example of boron sequence at high Z 3: the above considerations have been applied as an example to transitions between the ls22s22p and 1s22s2p2 configurations in the boron sequence. Elements of H,, were obtained by fitting an equation of the form [4]
each value of Z to obtain intermediate coupling eigenvalues and eigenvectors. Each eigenvector contains a single LS term for the ls22s22p configuration and up to three LS terms in the 1~22~2~2 configuration. A few sample eigenvectors are shown in table 1. Differences in the eigenvalues were used to calculate wavelengths.. Non-empirical LS multiplet strengths for each of ‘LheLS allowed transit&s between the ls22s22p and ls22s2p?- configurations were calculated for silicon using the new atomic structure theory of Sinanogu [I]. These LSJ multiplet strengths, which include the important “non-dynamical” correlation effects of NCMET, were converted to LSJ line strengths and extrapolated to high Z’s with equations of the Form s (,Z) = z(zo) Z$Z” . intermediate coupling line strengths were then calculated from these LSJ line strengths and the eigenvectors of eq. (1) by use of the equation
E(LS,Z) X K*(L
= c, (L,S) f [Z-Z, S) + [Z -Zg(L
(Ls)] S)l C,(L,
(2) 91
to empirical low Z data [S] (,Z=S to Z=lS) and extrapolating to high 2’s. Elements of HSO were obtained using the matrices given in Condon and Shortley [6] and spin-orbit coupling parameters obtained by fitting an equation of the form [4] t(Z) = Cso (Z-
Zso)”
(3)
to empirical low Z rirre structure data [S]. The parameters obtained are Cso = 0.246 cm-l and Zso -- ’_.& 192 for the ls22s22p configuration and Cgo = 0.248 cm-l and Z&, = 2.479 for the ls*2s2p2 configuration. HI, was then obtained from eq. (1) and diagonalized at 408 ‘.
are from this York, with intermediate coupling upper states composed
of P1,2, 2s 1~2, and ‘PI,2 terms.
= ~~~C(L2,S~;J,)S1n(L,, t ,
S,.J, -+L,,S,,
J2)
2
.
Volume 20, number
5
CHEhlICAL PHYSICS LETTERS
Table 1 Intermediate coupling eigcnvalues and eigenvectors states of the ls22s2p2 configuration”). P z=15 E(31 2PI:2 %/2 4P1,2
E(2) ZP1/2 2s112
4PI/2 E(1) 2P1/2 2.5,,2
4P1/2
hln z=25
Br z=35
for J=l/2
Table 2 Transition energies, A E, line strengths, S, and spontaneous emission rates, A, in intermediate coupling for 2sz2po+ . 2s2p’ transitions a)
Rh Z=45
427000 0.9794 0.20 18 0.0113
889000 0.7459 0.6654 -0.0304
1711000 0.5758 0.8045 -0.1457
3257000 0.4538 0.8496 -0.2681
403000 -0.2013 0.9789 -0.0347
800000 -0.6568 0.7271 -0.1997
1415000 0.7684 -0.4717 0.4326
2538000 0.7984 -0.2538 0.5460
178000 -0.0180 0.0317 0.9993
366000 -0.1107 0.1689 0.9794
525000 -0.2793 0.3611 0.8897
4 32000 -0.3957 0.4623 0.7935
F Z=lS ‘P912 -+(3)1,2 AE 427000 0.197 s A 16 2EL2
AE s A
A
cigenvalues.
AE S
The Condon-Shortley phase convention was observed in taking the square roots. Once the wavelengths, X , and line slrengths, S, had been calculated, oscillator strengths, f. spontaneous transition probabilities, A, and lifetimes, 7, were calculated from them. Some AE’s, S’s and A’s are shown in table 2. These results do not in&de a!1 possible KL+KL’ transitions, but they represent the type and magnitude of the effects involved. Similar effects will also be found in KLM, etc., in-shell transitions. The f versus l/Z curves that result from these calculations are very different from earlier predictions [2]. In figs. I and 2, typical (non-relativistic) openshell correlation effects are seen at the low 2 ends of the curves. In fig. 1, as the 2Pj,2 and ‘D3/2 mix due to spin-orbit effects, one curve “steals intensity” from the other. An “avoided crossing” of the 2P1,2 and 2S,,? IeveIs causes the behaviour shown in tig. 2. Some of the curves in figs. 1 and 2 turn up sharply for 2 > 50 due to the rapidly increasing (2”) energy differences. Fig. 3 shows a transition corresponding to the SOAEl type. This curve goes to zero for low 2’s. As can be seen in table 1, LS terms mix so heavily for 2 > 30 that there is no longer any distinction between
+
&In Z=25
Br z=35
_Rh Z=45
889000
1711000 0.032 161
3257OOi) 0.016 5.50
1285000 0.0015 3.2
2OL9000 0.0022 18
1415000 0.079 23
‘538000 0.080 I32
526000 2.0 X f04 3 x 10-Z
4;zooo 3.6 x IO+ 3 x 10-Z
0.074 53
(3)1,2
417000 0.048
790000
4XlOd 0.002
3.5
2Pp,,(2)1!2 ‘ AE S
E(2), and E(3) represent the first, second, and third eigenvslues respectively, in units of cm-‘. The coefficients of each LS term in each eigenvector are given below the
‘) E(l),
I July 1973
403000 0.019 1.3
800000 0.039
2.0
2pP,2 +(1)1/z A
178000 4.3 x 10-S 2 x lo-’
366000 1.8 x 10-s 9 x Lo-4
a! Line strengths are in atomic units, A’s are in nsec-I_ Upper states nre numbered 1, 2, 3 in order of increasing energy for each value of the total angular momentum J.
0””
’
“I’
I
I
’
’
-:
T-* - -6 $ -8
i
-10
--__ _ III
m
45
15
l/Z
II
I
Hf3’ONc
I B
_3
Fig. 3. Boron isoelectronic sequence oscillator strengths for spin-orbit allowed electric dipolc (SOAEI) transitions versus 1/Z. The solid curve represents the 2s22p (J=3/2)+2s2pz (&l/2, +l) transition, and the broken curve represents the 2s22p (J=1/2)+2s2p2 (5=3/2, +1) transition. The low Z (Z 6 10) end of the solid curve corresponds ta the 2s22p ZP9,2 * 2s2p2 4P,,* transition, and the low 2 end of the broken curve corresponds to the 2s22p ‘PO3,2+2s2p2 4P3;2 Lransition. All values obtained from this work using intermediate coupling upper states containing 4P. 2D. 2P: and ‘1~’ terms.
409
Vo!umc 20, numbx
5
CHEMICAL PHYSICS LETTERS
El and SOAEi iransitions. In the method used here both types are treated as one and both may have comparable values. In addition to the spin-orbi: interaction there are other related effects such as the “spin-spin” and “spin-other-orbit” interactions [7,8]. However, these effects are small compared to the spin-orbit in neutral atoms, and they grow only as Z3 rather than Z4. Consequently they become less important in high 2 ions and they can be neglected in the current work. Even so, these effecrs only represent an expansion to first order of a Dirac-like hamiltonian in terms of c?Z2. Beyond Z=60, an expansion in a2Z2 can no longer be expected to be rapidly convergent, as assumed here. In addition, quantum field theoretic effects (“radiative corrections”) which come in as (QZ)~. finite nucleus effects, hyperfine interactions, etc., may become important. Because a relativistic N-electron hamiltonian which includes all of these effects has not yet been developed, it is impossibie to carry these calculations beyond Z z 60. Unlike in the non-relativistic case, it is currently impossible to predict the asymptotic behaviour of these calculations as Z approaches infinity. The accuracy of these calculations is expected to range from about 10% for Z= 15 to factor of two or SO at 2~50, with certain exceptions. The curves in figs. I and 2 are drawn out to Z=65 to illustrate the behaviour of this model, but these last points are likely td he seriously in error. High-Z-low-N ions such as the ones discussed here can be produced in the laboratory by passing a less highly ionized beam of atoms through a thin carbon foil. The beam velocity required can be calculated from the semi-empirical formula of Dmitriev and Nicolaev [9, lo]. The corresponding beam energies arc ob!ainable for many atoms on Tandem Van de Graaff accelerators and the “SUPER-HILAC” at Berkeley. The transitions of these ions can then be studied by the technique of beam-foil spectroscopy. provided the distances travelled by the excited ions fall within measurable limits, typically on the order of one meter > VT > one millimeter [ 1 I]. The velocities calculated from ref. [lo], and transition rates obtained using the method described above predict that inany tIansitions decay within the measurable limits. .indicating the feasibility of observing these ions in the laboratory. ’ Any knowledge that wou!d be gained by the study 410...
.. .. ..: :
,’
1 July 1973
of these high-Z-low-N ions would be useful in developing a relativistic many-electron theory. Such a theory would be useful in chemistry because (a) due to the orbital orthogonality, relativistic effects of inner shell electrons are passed on to outer electrons, makmgi such effects important in less highly ionized and neutral atoms. (b) Inatomic and molecular calculations where the variational principle is applied, one needs experimental total energies for comparison with calculations. These experimental energies, which are needed also in obtaining the “experimental correlation energies”, involve a sum over all the ionization potentials of the atom(s) involved, and for heavy atoms most of these have not been observed. The theory and experiments discussed here would help fdl in the gaps in such data. (c) In addition, Auger and CosterKronig pro:esses. ESCA, and X-ray photoelectron spectroscopy frequently involve trarlsitions in which the initial znd/or final states have open L-shells similar to the ones considered above. The properties such as transition probabilities, inner binding energies, ESCA. splittings, etc., will all be affected by the same effects . presented here. This work was supported by a grant for the U.S. National Science Foundation. We thank Professor J. Rasmussen and Dr. Donald R. Beck for helpful dis-
cilssions. References Ill 0. Sinano$lu, Atomic physics, Vol. 1 (Plenum Press, New York, 1969) p. 131; P. Westhaus and 0. SinanoHlu, Phys. Rev. 183 (1969) 45; 0. Sinanogu, NucI. Instr. .?lcthods, 3rd Beam Foil SpectrOsCOpy COnfCrCnCe ISSUC(1973). I21 W.L. Wiese and .4.W. \Veiss, Phys. Rev. 175 (1968) 50. 131 R.H. Gxstang, in: Topics in modern physics, eds. WJZ. Btittin and H. Odabasi (Colorado Assoc. Univ. Press, Boulder, 197 1). [41 B. Edle’n, in: Handbuch der Physik, Vol. 27, cd. S. F16ge (Springer, Berlin, 1964) p. 156. [51 C. hloore, Atomic energy 1~~~1s.Vol. 1, NBS Circular 467 (US Govt. Ptinting Office, Washington, 1949). [61 E.U. Condon and G.H. Shortley, The theory of atomic spectra (Cambridge Univ. Press, London, 1935). I71 M. Blunie and R.E. Watson, Proc. Roy. Sot. A270 (1962) 127; A271 (1963) 565. ial R.E. Trxs, Phys. Rev. 82 (1951) 683. 191 H.D. Betz, Rev. Mod. Phys. 44 (1972) 465. 1101 I.S. Dmitriev and V.S. Nikolaev, Phys. Letters 28A (1968) 277. 1111 R. Marrus and R_ Schmiider, Phys. Rev. A5 (1972) 1160.