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Parametric treatment of the 3dn−1 4p configurations in trebly ionized atoms
Physica 75 (1974) 341-350 © North-Holland Publishing Co.
P A R A M E T R I C T R E A T M E N T O F T H E 3d n - 1 4 p C O N F I G U R A T I O N S IN T R E B L Y I O N I Z E D A T O M S R.'POPPE
~
Zeeman-Laboratorium der Universiteit van Amsterdam, Amsterdam, Nederland
Received 15 March 1974
Synopsis Well-known relations between parameters describing the structures of adjacent atomic species of the same ionization stage are applied to the 3d n - 1 4p configurations in the trebly ionized atoms of the iron group (V IV-Ga IV) known from analysis of observed spectra. A set of 32 general parameters is found to describe 344 atomic levels in 8 spectra, with satisfactory precision, the mean error being I 13 cm-1. This result is reached only when a new interaction term ~2L(L + 1) is introduced in addition to the usual Trees correction ~q L'(L' + 1), L' being the orbital quantum number of the d n - 1 parent. It is shown that the values of ~t2 in the various spectra are very consistent and obey a linear relation (with general parameters ~ and A~t2) analogQus to the behaviour of ~t in a d n, or a d n - 1 s configuration. It is expected that the general parameter values found will be helpful in establishing new levels in spectra which are known insufficiently. 1. I n t r o d u c t i o n . In 1966 N o o r m a n and Schrijver 1,2) calculated the 3d n, 3d n - 1 4s and 3d n-1 4 p c o n f i g u r a t i o n s o f t h e f o u r t h spectra o f the irong r o u p elements, using m e t h o d s d e s c r i b e d b y R a c a h and S h a d m i 3) and Shadmi4,s). T h e n u m b e r o f e x p e r i m e n t a l levels in the 3d n - 1 4p configurations k n o w n f r o m analysis o f optical spectra was o n l y 122 at t h a t time. Since t h e i r work, and p a r t l y also with the help o f t h e i r p r e d i c t i o n s o f levels and spectra, great advance has b e e n m a d e in the analysis. A d d i t i o n a l levels have b e e n f o u n d or c o r r e c t i o n s have b e e n m a d e in the spectra Cr IV 6) and Mn IV 7) while e n t i r e l y new analyses are n o w available f o r Fe IVS,9), Ni IV l°,aa) and Cu IVl~). T h e a u t h o r a n a l y z e d Co IV and i d e n t i f i e d s o m e low lying levels. So f o r n e w calculations o f t h e d n - 1 p c o n f i g u r a t i o n s , we c o u l d use 4 4 6 o f the 9 0 8 possible levels o f the f o u r t h spectra o f V ( Z = 23) u p to Ga ( Z = 31). T h e p r i m a r y goals o f this investigation have b e e n to get m o r e reliable p r e d i c t i o n s f o r t h e h i t h e r t o u n k n o w n h i g h e r levels, and t o o b t a i n a b e t t e r d e s c r i p t i o n o f the o n l y v e r y partially a n a l y z e d Co IV s p e c t r u m .
341
342
R. POPPE
A further purpose was to see whether the set o f parameters used by Noorman and Schrijver for the description o f the d n - l p configurations would remain adequate after the large increase o f the n u m b e r of known levels. 2. General m e t h o d o f calculation. It has been shown that parameters describing configurations in a sequence o f elements in one and the same state o f ionization are related in a rather simple way. This has been used for the sequence V I V - G a IV by Noorman and Schrijver 1,2) in general least-squares (GLS) calculations. The relations between the values of a parameter throughout the sequence (linear, quadratic or cubic) with coefficients determined by a running n u m b e r n, have been shown empirically, in several cases, to give satisfactory results. The GLS m e t h o d is superior to the m e t h o d o f independent least-squares (ILS) calculations in single spectra, if the configurations concerned are quite incomplete because erratic parameter values due to individual perturbations are m u c h less likely to occur. This applied especially to the d n - 1 p configurations. In the present situation, however, the GLS m e t h o d has few advantages. Most of the spectra are well analyzed now, so that for each single d n - 1 p configuration a comparatively reliable set of parameter values can be derived. This includes the parameter Ear which can be simply obtained as the centre o f gravity o f the configuration instead of being approximated from that o f the d n configuration 2 ). However, since Ear is merely an additive constant for all levels in one configuration it is o f no interest for the internal structure. Therefore it seemed appropriate to make independent least-squares calculations first (ILS-I), and then try to refine the parameter values obtained, except Ear, in a parameter least-squares (PLS) fit to the polynomial P = ff + (n - 6) A l P + [(n - 6) 2 - 10] A2P + (n - 6) 3 A3P o f suitable degree. Finally the set o f parameter values obtained from the i~LS calculation can be used in a second independent least-squares (ILS-II) fit with Ear as the only free parameter in order to minimize the error in the level positions measured with respect to the ground state. The procedure described avoids the calculation of matrices of unwieldy size. As will be shown it leads to very satisfactory results, especially after introducing a new parameter closely related to the usual Trees correction. 3. Results. The first ILS calculations, for some members of the group, started with the results given by Noorman and Schrijver; the parameters,
PARAMETRIC TREATMENT OF 3d n- 1 4p IN IV-SPECTRA
343
e x c e p t p a n d T, were then allowed to vary. A closer comparison o f the best values with the actual values o f the levels subsequently • revealed a systematic behaviour o f the (rather small) discrepancies that suggested a dependence on the total orbital q u a n t u m n u m b e r L, which is not described b y the Trees parameter a. The parent terms are split up into triads by the introduction o f the p electron. It is in each of those triads that residual discrepancies are seen which tend to cancel each other. In view o f this a new term a 2 L ( L + 1) was tried; it is obvious that such an effect cannot exist in a dn-ls configuration. The new approach proved to be a striking success, as can be seen from fig.l, representing the results for 3d24p of CrIV. The left-hand scale applies to the actual level positions whilst the departures of the calculated values from those actual positions are drawn on a scale blown up by a factor 20, as indicated by the arrow on the right-hand side. The corresponding level positions, calculated b y means o f the usual parameters, are given b y the horizontal lines ending in open circles. It is from such diagrams that it became clear that the departures are not randomly distributed; with some exceptions one finds that in a triad the higher-L term has a negative and the lower-L term has a positive departure. The level positions as calculated after introducing the new parameter ~t2 are given b y the horizontal lines ending in full circles; they are much closer to the actual positions generally. Much the same situation was found in the other spectra o f the group so that a real physical p h e n o m e n o n is suggested. This is borne out b y the fact that the ILS-I fits in all spectra o f the group except Cu IV give a very consistent set of values for at2. A survey o f the parameter values in units o f wavenumber, resulting from the ILS-I calculations, b o t h with and without ~2 is presented in table I. In each block the upper n u m b e r is the value obtained when using ot1 and ot2, and the lower one is the result o f the best fit without ~2. Because # and T were kept fixed they are not shown in the table; their values are " 3 0 0 , respectively - 5 c m - i in all cases. Most significant is the reduction in the mean error a, defined by a = [(~ir~)/(n-
m)l ½,
where 6 i is the difference b e t w e e n the experimental and the calculated value o f a level, n is the n u m b e r o f levels used and rn the n u m b e r of free parameters. Because the n u m b e r o f free parameters is accounted for in a, the improvement is n o t due to the introduction o f just another parameter. In only one spectrum (Cu IV) no improvement at all is obtained. The last line in table I contains the values o f the mean error obtained in
344
R. POPPE
an ILS-II calculation, as described in section 2, again with and without =2. In the PLS procedure a first-degree polynomial was found to be satisfactory for F2(dd), F2(dp), Gl(dp), G3(dp) and =1, =2~ respectively =, whilst a second-degree term had to be taken into account for P ( d d ) , £d and (p. The general picture is the same as for tr from the ILS-I fits: significant improvements, except for Cu IV.
e'3U O ,¢210
(IS)2p--~ lS)2pj:~ 1100 cm-1
200
190
(1G)2FZ (3p)2p...~ 180
(3p)4p.._c=p~ (3p)4D__~ 170
(3p)2p~..o (3P)2D L4
(1G)2H"]~
-.~
(3p)4p ~ .-,-o (3p)4p ~ (~D)2DJ (1D)2D.J ('IP):D-'E~ (.D)4P -'1 ('D)-F-'1~ ('P) S-=E:
(1G)2GJ:~ (3p)4D
IG)2G.-[~
(3F)'2G ~
(3F)2G
(1G)2H"~
(3p)2s--I (3F)4DJ=~
160
150
(3p)2D
(1G)2FZ
3F)4F-%
1/2
3/2
( F#D Ir-o (3F)2F-. (3F)4F Le (3F)4G,,--E~
5/2
(~F')~F"-~ (3F)4F (3F)4F --1 (3F)4G~ "~ (3F)4G'~ (3F-)4G,,--[~I~
7/2
9/2
11/2
J
Fig.l. Actual level positions in Cr IV 3d24p drawn to scale (left) and departures of calculated level positions shown in a 20-fold expanded scale (fight). Open circles correspond to results of conventional calculations; full circles refer to results obtained after including the =2 correction terrrL
345
P A R A M E T R I C T R E A T M E N T OF 3d n - ~ 4 p IN I V - S P E C T R A
03
02
t'q
tt3
t'-q ,,~-
t'qoo
,-~
,...4
¢qc.q
t'q
,--q
oot~
,¢-v~
t~
oo
c4
O
o~
t'q
c4,~-
tg~
oo
o'~ t',t~
t~
tn~
o~o
v.,.q ,.-q
...4
t~-(q
L'-,- ,.-~
la
b
v.,,~
> tr~ ',O
tnt.q t"q t¢~
t'q
t.qtn
6"
346
R. POPPE
It should be n o t e d that the values a (ILS-II) do not correspond to the rest o f the table. The complete set o f parameters used in the final calculation is recorded in the first column of table II. In the second column are the values as given b y Noorman and Schrijver, except that the values o f Ear(n) have been deduced from the barycentres of the ground configurations and the values o f D, A1D, zt2D and A3D given in ref. 2, according to the formula for the distances o f the barycentres: m
D = D + (n - 6 ) A ~ D + [(n - 6) 2 - 10] A2D + (n - 6) 3 A3D.
No values for Ear (6) and Ear (7) were available because at that time the ground configurations were insufficiently known. In the next two columns are the results obtained in the present investigation, with and without a2 successively. Because there are serious doubts with regard to the present analysis of Cu IV and because the Cu IV data have an adverse effect on the mean error, as mentioned above, Cu IV was omitted from the PLS fit. Co IV was not used either, because the number of known levels is insufficient to perform a reliable ILS-I calculation. A typical feature connected with the introduction of the a2 correction, which can be seen from table II as well as table I, is the reduction which it brings a b o u t in the value of • (which becomes ~ ). Obviously part of the conventional Trees correction is absorbed by the "second generation" Trees correction. Actually a 1 and ~2 turn out to be comparable in magnitude; their sums are about equal to the conventional parameter found if ~2 is assumed to be zero, as can be expected. Complete information on the mean-error evaluation is collected in table III. The cases A, B and C correspond to the parameter values in the successive columns of table II, that is those obtained when Cu IV is included, while A', B', and C' are the analogous cases with Cu IV excluded. F r o m the last column, giving the mean error, it is seen that the exclusion of the 102 Cu IV levels produces a definite improvement in all cases; however, in case C/C' one does n o t only find the most spectacular improvement, b u t also the smallest absolute values. This can be regarded as a further indication of the physical significance o f the ~2 correction. 4. Discussion. It has already been remarked in section 3 that the Cu IV
data are disturbing elements in the calculations although not sufficiently disturbing to mask the beneficial effect o f introducing ~'2- At the same time it shows the sensitivity of the m e t h o d o f computing general parameters to detect irregularities in a particular spectrum, provided most of the spectra used are correctly analyzed. In the case of Cu IV it should be mentioned that a couple o f high levels were discarded from the original
347
PARAMETRIC TREATMENT OF 3d n - 1 4p IN IV-SPECTRA
TABLE II General parameters for the configurations 3d n - 14p Parameter
Ear Ear Ear Ear Ear Eav Ear Ear Ear
(2) (3) (4) (5) (6) (7) (8) (9) (10)
F2(dd) 4~___F2(dd) F4(dd) AiF4(dd) A2F4(dd) ~d
AI~d d2Cd ~'p Al~p A_~2¢p F2(dp) A__3F~(dp) G 1 (dp) dlGX(dp) Ga(dp) zl~ G 3 (dp) a-~
di~'i a-~ dl~ 2 P(fixed) T (fixed)
Ref. 2 148 890 170 707 194 436 228 639 222 435 221 468 231 258 93 820 4 547 59 800 3 704 0 (fixed) 766 144 14 1 091 121 8 21 283 360 7 390 35 6 313 172 54.0 -0.3 0 (fixed) 0 (fixed) -300 -5
ILS-II
ILS-II
without ~'2
with ~'2
148 170 194 228 221 222 222 221 232 93 4 60 3
859 770 288 432 123 236 437 916 357
738 541 474 545 101 771 144 11 1 094 127 4 21 424 330 7 405 39 6 238 169 55.5 -0.1 0 (fixed) 0 (fixed) - 300 -5
148 170 194 228 221 222 222 221 232
851 763 278 399 106 203 378 851 293
93 4 60 3
700 510 395 548 93 776 148 12 1 130 133 5 21 402 306 7 644 11 5 804 112 28.2 1.5 27.0 -0.5 -300 -5
ILS-I calculation but this gave no improvement. The conclusion therefore is that something more fundamental is wrong in the existing analysis~2). This confirms objections raised by Professor Shenstone in a private conversation (Klinkenberg, Zaragoza, September 1972); for this reason the analysis is now under revision by Meinders. In accordance with these considerations the Cu IV parameters have been left out of the final PLS calculations.
348
R. POPPE TABLE III
Mean errors Case
(104~6~ cm_2)
n
rn
n- m
a
A A' B B' C C'
1117 715 1039 615 873 367
446 344 446 344 446 344
27 26 28 27 30 29
419 318 418 317 416 315
163 150 158 140 145 113
Looking for other irregularities o f a similar nature it is seen from table I that in the ILS-I calculations w i t h o u t ~t2 nearly all parameters of V IV 3d4p show irregular behaviour. F o r that reason this system was not used in the corresponding PLS approximation. This had very little influence on the ultimate mean deviation in V IV; here the "deviation" is taken to be the root mean-square value of the difference between the calculated and experimental levels. The biggest irregularity is in (p whose value 1139 is far too high also in connection with the preceding spectrum, Ti IV, which is a one-electron system. Remarkably enough this high figure is supported by a recent study o f the Sc II iso-electronic sequence13), leading to a value 1032 for ( p ( V IV 3d4p). It is seen from table I that after the introduction of ~t2 most irregularities in V IV disappear. In particular (p fits perfectly in the sequence of values for the whole group. An extrapolation to Ti IV now gives (p (Ti I V ) = 579 while the actual value is 545 (from the 2p interval of 818 cm-1). The conclusion is that even in configurations as simple as dp the neglect of a2 can cause significant departures in the calculated values of ~p and possibly of other parameters too. The fact that this effect is not detected in an iso-electronic treatment is not astonishing; because the spectra concerned are so much alike this effect is easily absorbed, especially in the values. The superiority o f the comparison of spectra of the same ionization in this respect is due to the circumstance that the perturbations work out differently in the successive spectra. Hence one should expect discrepancies between the results of the two treatments, when a2 is neglected. An indication o f this kind of discrepancy is found in Ti III 3d4p, dealt with in ref. 13 and in a previous study o f 3dn-14p configurations in doubly ionized atoms, using a GLS approximation14). This gives ~p = 328, while
PARAMETRIC TREATMENT OF 3dn- 1 4p IN IV-SPECTRA
349
ref. 13 leads t o ~p = 487. The GLS approximation in the III spectra comprised 581 levels and the mean error obtained was 138, which is approximately the same as found in our study of the IV spectra without ~t2. A detailed inspection o f the deviations in the former case reveals a systematic L dependence so that an improvement should be expected in that calculation also upon introduction o f the ~2 correction. Returning finally to the problem o f Co IV it is unrealistic to expect that mean errors as given in tables I and III are a reliable estimate o f the uncertainty in the prediction o f unknown levels. As an illustration of this it is worthwhile to examine the case of Cr IV 3d 2 4p. In ref. 1 there were 24 accepted levels and a fit was obtained giving a mean error of 111 cm -1, in accordance with the general value for 3dn-q~ 4p which was 114 cm -I . The mean deviation for the GLS calculation of NQorman and Schrijver for those levels is 117 c m -1. However, if the complete configuration (45 levels) as is now k n o w n 6) is included, this figure becomes 171 cm -1, and for the new levels alone it is 212 cm -1. In our final ILS-II calculation, making use o f the complete set o f levels and including the ~t: correction, the mean deviation for Cr IV 3d 2 4p is reduced to 78 cm -~ which is better than before. Therefore, if we believe the new predictions for Co IV to be more accurate than previous ones, this is not only based on the improvement of the description but also on the fact that far more k n o w n levels and more complete configurations could be taken into account. Acknowledgements. The author expresses his indebtedness to Professor B. Edl6n for making available his new results in the analysis of Fe IV. The author wishes to thank Dr. E. Meinders for discussions during the work, her and Mr. I. A. D. Bruinvis for help with the computer programs. Mr. H. Schans for drawing the figure, Dr. P. F. A. Klinkenberg for improving the manuscript and Dr. J. Schrijver for discussions about the physical significance o f the parameters. The o p p o r t u n i t y to use the CDC 6400 computer of the Stichting voor F u n d a m e n t e e l Onderzoek der Materie at the Zeeman Laboratorium was highly appreciated.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8)
Schrijver, J. and Noorman, P. E., Physica 32 (1966) 357. Noorman, P. E. and Schrijver, J., Physica 36 (1967) 547. Racah, G. and Shadmi, Y., Bull. Res. Counc. Israel 8F (1959) 15. Shadmi, Y., Bull. Res. Counc. Israel 10F (1962) 109. Shadmi, Y., Phys. Rev. 139 (1965) A43. Ekberg, J. O., Physica Scripta 7 (1973) 55. Yarosewick, S. J. and Moore Jr., F. L., J. Opt. Soc. Amer. 57 (1967) 1381. Edl~n, B., Mon. Not. R. astr. Soc. 144 (1969) 391.
350 9) 10) 11) 12) 13) 14)
R. POPPE Edl~n, B., private communication. Poppe, R., Physica 40 (1968) 17. Garc~a-Riquelme, Olga, Physica 40 (1968) 27. Schr~der, J. F. and Van Kleef, Th.A.M., Physica 49 (1970) 388. Warner, B. and Kirkpatrick, R. C., Mon. Not. R. astr. Soc. 144 (1969) 397. Roth, C., J. Res. Nat. Bur. Stand. 72A (1968) 505.
P A R A M E T R I C T R E A T M E N T O F T H E 3d n - 1 4 p C O N F I G U R A T I O N S IN T R E B L Y I O N I Z E D A T O M S R.'POPPE
~
Zeeman-Laboratorium der Universiteit van Amsterdam, Amsterdam, Nederland
Received 15 March 1974
Synopsis Well-known relations between parameters describing the structures of adjacent atomic species of the same ionization stage are applied to the 3d n - 1 4p configurations in the trebly ionized atoms of the iron group (V IV-Ga IV) known from analysis of observed spectra. A set of 32 general parameters is found to describe 344 atomic levels in 8 spectra, with satisfactory precision, the mean error being I 13 cm-1. This result is reached only when a new interaction term ~2L(L + 1) is introduced in addition to the usual Trees correction ~q L'(L' + 1), L' being the orbital quantum number of the d n - 1 parent. It is shown that the values of ~t2 in the various spectra are very consistent and obey a linear relation (with general parameters ~ and A~t2) analogQus to the behaviour of ~t in a d n, or a d n - 1 s configuration. It is expected that the general parameter values found will be helpful in establishing new levels in spectra which are known insufficiently. 1. I n t r o d u c t i o n . In 1966 N o o r m a n and Schrijver 1,2) calculated the 3d n, 3d n - 1 4s and 3d n-1 4 p c o n f i g u r a t i o n s o f t h e f o u r t h spectra o f the irong r o u p elements, using m e t h o d s d e s c r i b e d b y R a c a h and S h a d m i 3) and Shadmi4,s). T h e n u m b e r o f e x p e r i m e n t a l levels in the 3d n - 1 4p configurations k n o w n f r o m analysis o f optical spectra was o n l y 122 at t h a t time. Since t h e i r work, and p a r t l y also with the help o f t h e i r p r e d i c t i o n s o f levels and spectra, great advance has b e e n m a d e in the analysis. A d d i t i o n a l levels have b e e n f o u n d or c o r r e c t i o n s have b e e n m a d e in the spectra Cr IV 6) and Mn IV 7) while e n t i r e l y new analyses are n o w available f o r Fe IVS,9), Ni IV l°,aa) and Cu IVl~). T h e a u t h o r a n a l y z e d Co IV and i d e n t i f i e d s o m e low lying levels. So f o r n e w calculations o f t h e d n - 1 p c o n f i g u r a t i o n s , we c o u l d use 4 4 6 o f the 9 0 8 possible levels o f the f o u r t h spectra o f V ( Z = 23) u p to Ga ( Z = 31). T h e p r i m a r y goals o f this investigation have b e e n to get m o r e reliable p r e d i c t i o n s f o r t h e h i t h e r t o u n k n o w n h i g h e r levels, and t o o b t a i n a b e t t e r d e s c r i p t i o n o f the o n l y v e r y partially a n a l y z e d Co IV s p e c t r u m .
341
342
R. POPPE
A further purpose was to see whether the set o f parameters used by Noorman and Schrijver for the description o f the d n - l p configurations would remain adequate after the large increase o f the n u m b e r of known levels. 2. General m e t h o d o f calculation. It has been shown that parameters describing configurations in a sequence o f elements in one and the same state o f ionization are related in a rather simple way. This has been used for the sequence V I V - G a IV by Noorman and Schrijver 1,2) in general least-squares (GLS) calculations. The relations between the values of a parameter throughout the sequence (linear, quadratic or cubic) with coefficients determined by a running n u m b e r n, have been shown empirically, in several cases, to give satisfactory results. The GLS m e t h o d is superior to the m e t h o d o f independent least-squares (ILS) calculations in single spectra, if the configurations concerned are quite incomplete because erratic parameter values due to individual perturbations are m u c h less likely to occur. This applied especially to the d n - 1 p configurations. In the present situation, however, the GLS m e t h o d has few advantages. Most of the spectra are well analyzed now, so that for each single d n - 1 p configuration a comparatively reliable set of parameter values can be derived. This includes the parameter Ear which can be simply obtained as the centre o f gravity o f the configuration instead of being approximated from that o f the d n configuration 2 ). However, since Ear is merely an additive constant for all levels in one configuration it is o f no interest for the internal structure. Therefore it seemed appropriate to make independent least-squares calculations first (ILS-I), and then try to refine the parameter values obtained, except Ear, in a parameter least-squares (PLS) fit to the polynomial P = ff + (n - 6) A l P + [(n - 6) 2 - 10] A2P + (n - 6) 3 A3P o f suitable degree. Finally the set o f parameter values obtained from the i~LS calculation can be used in a second independent least-squares (ILS-II) fit with Ear as the only free parameter in order to minimize the error in the level positions measured with respect to the ground state. The procedure described avoids the calculation of matrices of unwieldy size. As will be shown it leads to very satisfactory results, especially after introducing a new parameter closely related to the usual Trees correction. 3. Results. The first ILS calculations, for some members of the group, started with the results given by Noorman and Schrijver; the parameters,
PARAMETRIC TREATMENT OF 3d n- 1 4p IN IV-SPECTRA
343
e x c e p t p a n d T, were then allowed to vary. A closer comparison o f the best values with the actual values o f the levels subsequently • revealed a systematic behaviour o f the (rather small) discrepancies that suggested a dependence on the total orbital q u a n t u m n u m b e r L, which is not described b y the Trees parameter a. The parent terms are split up into triads by the introduction o f the p electron. It is in each of those triads that residual discrepancies are seen which tend to cancel each other. In view o f this a new term a 2 L ( L + 1) was tried; it is obvious that such an effect cannot exist in a dn-ls configuration. The new approach proved to be a striking success, as can be seen from fig.l, representing the results for 3d24p of CrIV. The left-hand scale applies to the actual level positions whilst the departures of the calculated values from those actual positions are drawn on a scale blown up by a factor 20, as indicated by the arrow on the right-hand side. The corresponding level positions, calculated b y means o f the usual parameters, are given b y the horizontal lines ending in open circles. It is from such diagrams that it became clear that the departures are not randomly distributed; with some exceptions one finds that in a triad the higher-L term has a negative and the lower-L term has a positive departure. The level positions as calculated after introducing the new parameter ~t2 are given b y the horizontal lines ending in full circles; they are much closer to the actual positions generally. Much the same situation was found in the other spectra o f the group so that a real physical p h e n o m e n o n is suggested. This is borne out b y the fact that the ILS-I fits in all spectra o f the group except Cu IV give a very consistent set of values for at2. A survey o f the parameter values in units o f wavenumber, resulting from the ILS-I calculations, b o t h with and without ~2 is presented in table I. In each block the upper n u m b e r is the value obtained when using ot1 and ot2, and the lower one is the result o f the best fit without ~2. Because # and T were kept fixed they are not shown in the table; their values are " 3 0 0 , respectively - 5 c m - i in all cases. Most significant is the reduction in the mean error a, defined by a = [(~ir~)/(n-
m)l ½,
where 6 i is the difference b e t w e e n the experimental and the calculated value o f a level, n is the n u m b e r o f levels used and rn the n u m b e r of free parameters. Because the n u m b e r o f free parameters is accounted for in a, the improvement is n o t due to the introduction o f just another parameter. In only one spectrum (Cu IV) no improvement at all is obtained. The last line in table I contains the values o f the mean error obtained in
344
R. POPPE
an ILS-II calculation, as described in section 2, again with and without =2. In the PLS procedure a first-degree polynomial was found to be satisfactory for F2(dd), F2(dp), Gl(dp), G3(dp) and =1, =2~ respectively =, whilst a second-degree term had to be taken into account for P ( d d ) , £d and (p. The general picture is the same as for tr from the ILS-I fits: significant improvements, except for Cu IV.
e'3U O ,¢210
(IS)2p--~ lS)2pj:~ 1100 cm-1
200
190
(1G)2FZ (3p)2p...~ 180
(3p)4p.._c=p~ (3p)4D__~ 170
(3p)2p~..o (3P)2D L4
(1G)2H"]~
-.~
(3p)4p ~ .-,-o (3p)4p ~ (~D)2DJ (1D)2D.J ('IP):D-'E~ (.D)4P -'1 ('D)-F-'1~ ('P) S-=E:
(1G)2GJ:~ (3p)4D
IG)2G.-[~
(3F)'2G ~
(3F)2G
(1G)2H"~
(3p)2s--I (3F)4DJ=~
160
150
(3p)2D
(1G)2FZ
3F)4F-%
1/2
3/2
( F#D Ir-o (3F)2F-. (3F)4F Le (3F)4G,,--E~
5/2
(~F')~F"-~ (3F)4F (3F)4F --1 (3F)4G~ "~ (3F)4G'~ (3F-)4G,,--[~I~
7/2
9/2
11/2
J
Fig.l. Actual level positions in Cr IV 3d24p drawn to scale (left) and departures of calculated level positions shown in a 20-fold expanded scale (fight). Open circles correspond to results of conventional calculations; full circles refer to results obtained after including the =2 correction terrrL
345
P A R A M E T R I C T R E A T M E N T OF 3d n - ~ 4 p IN I V - S P E C T R A
03
02
t'q
tt3
t'-q ,,~-
t'qoo
,-~
,...4
¢qc.q
t'q
,--q
oot~
,¢-v~
t~
oo
c4
O
o~
t'q
c4,~-
tg~
oo
o'~ t',t~
t~
tn~
o~o
v.,.q ,.-q
...4
t~-(q
L'-,- ,.-~
la
b
v.,,~
> tr~ ',O
tnt.q t"q t¢~
t'q
t.qtn
6"
346
R. POPPE
It should be n o t e d that the values a (ILS-II) do not correspond to the rest o f the table. The complete set o f parameters used in the final calculation is recorded in the first column of table II. In the second column are the values as given b y Noorman and Schrijver, except that the values o f Ear(n) have been deduced from the barycentres of the ground configurations and the values o f D, A1D, zt2D and A3D given in ref. 2, according to the formula for the distances o f the barycentres: m
D = D + (n - 6 ) A ~ D + [(n - 6) 2 - 10] A2D + (n - 6) 3 A3D.
No values for Ear (6) and Ear (7) were available because at that time the ground configurations were insufficiently known. In the next two columns are the results obtained in the present investigation, with and without a2 successively. Because there are serious doubts with regard to the present analysis of Cu IV and because the Cu IV data have an adverse effect on the mean error, as mentioned above, Cu IV was omitted from the PLS fit. Co IV was not used either, because the number of known levels is insufficient to perform a reliable ILS-I calculation. A typical feature connected with the introduction of the a2 correction, which can be seen from table II as well as table I, is the reduction which it brings a b o u t in the value of • (which becomes ~ ). Obviously part of the conventional Trees correction is absorbed by the "second generation" Trees correction. Actually a 1 and ~2 turn out to be comparable in magnitude; their sums are about equal to the conventional parameter found if ~2 is assumed to be zero, as can be expected. Complete information on the mean-error evaluation is collected in table III. The cases A, B and C correspond to the parameter values in the successive columns of table II, that is those obtained when Cu IV is included, while A', B', and C' are the analogous cases with Cu IV excluded. F r o m the last column, giving the mean error, it is seen that the exclusion of the 102 Cu IV levels produces a definite improvement in all cases; however, in case C/C' one does n o t only find the most spectacular improvement, b u t also the smallest absolute values. This can be regarded as a further indication of the physical significance o f the ~2 correction. 4. Discussion. It has already been remarked in section 3 that the Cu IV
data are disturbing elements in the calculations although not sufficiently disturbing to mask the beneficial effect o f introducing ~'2- At the same time it shows the sensitivity of the m e t h o d o f computing general parameters to detect irregularities in a particular spectrum, provided most of the spectra used are correctly analyzed. In the case of Cu IV it should be mentioned that a couple o f high levels were discarded from the original
347
PARAMETRIC TREATMENT OF 3d n - 1 4p IN IV-SPECTRA
TABLE II General parameters for the configurations 3d n - 14p Parameter
Ear Ear Ear Ear Ear Eav Ear Ear Ear
(2) (3) (4) (5) (6) (7) (8) (9) (10)
F2(dd) 4~___F2(dd) F4(dd) AiF4(dd) A2F4(dd) ~d
AI~d d2Cd ~'p Al~p A_~2¢p F2(dp) A__3F~(dp) G 1 (dp) dlGX(dp) Ga(dp) zl~ G 3 (dp) a-~
di~'i a-~ dl~ 2 P(fixed) T (fixed)
Ref. 2 148 890 170 707 194 436 228 639 222 435 221 468 231 258 93 820 4 547 59 800 3 704 0 (fixed) 766 144 14 1 091 121 8 21 283 360 7 390 35 6 313 172 54.0 -0.3 0 (fixed) 0 (fixed) -300 -5
ILS-II
ILS-II
without ~'2
with ~'2
148 170 194 228 221 222 222 221 232 93 4 60 3
859 770 288 432 123 236 437 916 357
738 541 474 545 101 771 144 11 1 094 127 4 21 424 330 7 405 39 6 238 169 55.5 -0.1 0 (fixed) 0 (fixed) - 300 -5
148 170 194 228 221 222 222 221 232
851 763 278 399 106 203 378 851 293
93 4 60 3
700 510 395 548 93 776 148 12 1 130 133 5 21 402 306 7 644 11 5 804 112 28.2 1.5 27.0 -0.5 -300 -5
ILS-I calculation but this gave no improvement. The conclusion therefore is that something more fundamental is wrong in the existing analysis~2). This confirms objections raised by Professor Shenstone in a private conversation (Klinkenberg, Zaragoza, September 1972); for this reason the analysis is now under revision by Meinders. In accordance with these considerations the Cu IV parameters have been left out of the final PLS calculations.
348
R. POPPE TABLE III
Mean errors Case
(104~6~ cm_2)
n
rn
n- m
a
A A' B B' C C'
1117 715 1039 615 873 367
446 344 446 344 446 344
27 26 28 27 30 29
419 318 418 317 416 315
163 150 158 140 145 113
Looking for other irregularities o f a similar nature it is seen from table I that in the ILS-I calculations w i t h o u t ~t2 nearly all parameters of V IV 3d4p show irregular behaviour. F o r that reason this system was not used in the corresponding PLS approximation. This had very little influence on the ultimate mean deviation in V IV; here the "deviation" is taken to be the root mean-square value of the difference between the calculated and experimental levels. The biggest irregularity is in (p whose value 1139 is far too high also in connection with the preceding spectrum, Ti IV, which is a one-electron system. Remarkably enough this high figure is supported by a recent study o f the Sc II iso-electronic sequence13), leading to a value 1032 for ( p ( V IV 3d4p). It is seen from table I that after the introduction of ~t2 most irregularities in V IV disappear. In particular (p fits perfectly in the sequence of values for the whole group. An extrapolation to Ti IV now gives (p (Ti I V ) = 579 while the actual value is 545 (from the 2p interval of 818 cm-1). The conclusion is that even in configurations as simple as dp the neglect of a2 can cause significant departures in the calculated values of ~p and possibly of other parameters too. The fact that this effect is not detected in an iso-electronic treatment is not astonishing; because the spectra concerned are so much alike this effect is easily absorbed, especially in the values. The superiority o f the comparison of spectra of the same ionization in this respect is due to the circumstance that the perturbations work out differently in the successive spectra. Hence one should expect discrepancies between the results of the two treatments, when a2 is neglected. An indication o f this kind of discrepancy is found in Ti III 3d4p, dealt with in ref. 13 and in a previous study o f 3dn-14p configurations in doubly ionized atoms, using a GLS approximation14). This gives ~p = 328, while
PARAMETRIC TREATMENT OF 3dn- 1 4p IN IV-SPECTRA
349
ref. 13 leads t o ~p = 487. The GLS approximation in the III spectra comprised 581 levels and the mean error obtained was 138, which is approximately the same as found in our study of the IV spectra without ~t2. A detailed inspection o f the deviations in the former case reveals a systematic L dependence so that an improvement should be expected in that calculation also upon introduction o f the ~2 correction. Returning finally to the problem o f Co IV it is unrealistic to expect that mean errors as given in tables I and III are a reliable estimate o f the uncertainty in the prediction o f unknown levels. As an illustration of this it is worthwhile to examine the case of Cr IV 3d 2 4p. In ref. 1 there were 24 accepted levels and a fit was obtained giving a mean error of 111 cm -1, in accordance with the general value for 3dn-q~ 4p which was 114 cm -I . The mean deviation for the GLS calculation of NQorman and Schrijver for those levels is 117 c m -1. However, if the complete configuration (45 levels) as is now k n o w n 6) is included, this figure becomes 171 cm -1, and for the new levels alone it is 212 cm -1. In our final ILS-II calculation, making use o f the complete set o f levels and including the ~t: correction, the mean deviation for Cr IV 3d 2 4p is reduced to 78 cm -~ which is better than before. Therefore, if we believe the new predictions for Co IV to be more accurate than previous ones, this is not only based on the improvement of the description but also on the fact that far more k n o w n levels and more complete configurations could be taken into account. Acknowledgements. The author expresses his indebtedness to Professor B. Edl6n for making available his new results in the analysis of Fe IV. The author wishes to thank Dr. E. Meinders for discussions during the work, her and Mr. I. A. D. Bruinvis for help with the computer programs. Mr. H. Schans for drawing the figure, Dr. P. F. A. Klinkenberg for improving the manuscript and Dr. J. Schrijver for discussions about the physical significance o f the parameters. The o p p o r t u n i t y to use the CDC 6400 computer of the Stichting voor F u n d a m e n t e e l Onderzoek der Materie at the Zeeman Laboratorium was highly appreciated.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8)
Schrijver, J. and Noorman, P. E., Physica 32 (1966) 357. Noorman, P. E. and Schrijver, J., Physica 36 (1967) 547. Racah, G. and Shadmi, Y., Bull. Res. Counc. Israel 8F (1959) 15. Shadmi, Y., Bull. Res. Counc. Israel 10F (1962) 109. Shadmi, Y., Phys. Rev. 139 (1965) A43. Ekberg, J. O., Physica Scripta 7 (1973) 55. Yarosewick, S. J. and Moore Jr., F. L., J. Opt. Soc. Amer. 57 (1967) 1381. Edl~n, B., Mon. Not. R. astr. Soc. 144 (1969) 391.
350 9) 10) 11) 12) 13) 14)
R. POPPE Edl~n, B., private communication. Poppe, R., Physica 40 (1968) 17. Garc~a-Riquelme, Olga, Physica 40 (1968) 27. Schr~der, J. F. and Van Kleef, Th.A.M., Physica 49 (1970) 388. Warner, B. and Kirkpatrick, R. C., Mon. Not. R. astr. Soc. 144 (1969) 397. Roth, C., J. Res. Nat. Bur. Stand. 72A (1968) 505.