ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY K EMISSION OF THE 3 d AND 4 d ELEMENTS

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ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY K EMISSION OF THE 3 d AND 4 d

ELEMENTS

J. Finster, G. Leonhardt, A. Meisel

To cite this version:

J. Finster, G. Leonhardt, A. Meisel. ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY K EMISSION OF THE 3 d AND 4 d ELEMENTS. Journal de Physique Colloques, 1971, 32 (C4), pp.C4-218-C4-224. �10.1051/jphyscol:1971441�. �jpa-00214642�

JOURNAL DE PHYSIQUE Colloque C 4 , supplkment au no 10, Tome 32, Octobre 1971, page C4-218

ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY K EMISSION OF THE 3 d AND 4 d ELEMENTS

J. FINSTER, G. LEONHARDT and A. MEISEL Sektion Chemie der Karl-Marx-Universitat Leipzig

RQum6. - Cet article traite de la modification du profil de raie (en particulier en ce qui concerne la largeur et l'asymetrie) se produisant pour les principales raies d'kmission de la serie K (Kal.2 et ^Kfl1.3). I1 expose et analyse les resultats de mesures obtenus jusqu'8 present. I1 presente des donnks originales pour les modifications de la forme et de la largeur du doublet Kal.2 de 42Mo

sous l'influence de la liaison chimique. La notion de section efficace de raie est introduite en tant que representation adequate de telles modifications. On tente d'ttablir une relation quantitative avec le nombre des 6lectrons non appariks pour les raies Koll des klkments 3 d.

L'interprktation la meilleure de ces effets est obtenue en appliquant la thkorie de la structure en multiplets des spectres de rayons X. Une analyse des calculs effectues de cette f a ~ o n montre la necessitk de delimiter clairement les degres d'approximation utilisks.

Ce travail traite notamment du mode de couplage des moments, en particulier du couplage ^yy.

Dans ce but les paramktres de couplage spin-orbite c 3 d et ^{5 4 6} sont determines pour les elements 3 d et 4 d respectivement par extrapolation. Nous remarquons que le couplage ^{y y} pur donne un Bargissement trop petit pour les raies des elements 3 d et trop grand pour celles des elements 4 d, et que la prise en considQation de l'interaction Blectrostatique (couplage intermediaire) am6liore l'accord avec l'experience dans le cas du doublet Kal.2 des elements 3 d (a I'oppose du cas des 616ments 4 d).

Abstract. - This paper deals with the changes of the line shape (in particular width and asym- metry) occuring with the main emission lines of the K series (Kol1.2 and Kb1.3). The measuring results gained so far are reviewed and analyzed. A first report is given on the changes of the shape and width of the Kal.2 doublet of 42Mo influenced by the chemical bond. As a relevant kind of representation of such changes the so-called line-cross section is introduced. For the Kal lines of the 3 d elements an attempt is made to give a quantitative relation to the number of unpaired electrons.

Most successful for the interpretation of these effects proved the application of the theory of the multiplet structure on X-ray spectra. An analysis of the calculations basing on this procedure reveals the necessity of a distinct limiting of the approximation stages used.

This work treats notably the mode of the coupling of the moments, in particular the jj-coupling.

For this purpose, the spin-orbit coupling parameters 5 3 d and 54d. resp., for 3 d and 4 d elements were determined by extrapolation.

We notice that the pure jj-coupling yields for the lines of the 3 d elements a too small broa- dening and for those of the 4 d elements a too large broadening and that for the Ka1.2 doublet of the 3 d elements (in contrast to that of the 4 d elements) an additional consideration of the electrostatic interaction (intermediate coupling) improves the agreement with experiment.

1. Introduction. - The study of the position of the main emission lines (i. e. in particular Ka,., with Z > 20 and KP, ., with Z > 35) and its change (chemical shift) has advanced experimentally and theoretically such as to provide a rather good picture (Meisel and coworkers [I], Sumbayev et al. [2];

relations with effective atomic charge [3,4]). The problems to be solved now are the transition from the model of the free atom or ion used so far to the solid and molecule, respectively.

The investigations of the shape and width of X-ray emission lines being of utmost significance for the theory of X-ray spectra as well as for the application of X-ray spectroscopy t o problems of chemical bond

and others are still beyond satisfaction. However, in recent years a large number of measurements

-

most with Ka,., doublets of 3 d metals and their compounds [5, 1, 6, 71 - and first calculations for the quantitative interpretation of effects 18-12] ^(I) have been carried out and offer a possibility for a better understanding of the situation.

This work is meant to give a contribution to this aim by examination, analysis and systematization of own results and those known from literature.

2. Experimental results. - 2.1 CONDITIONS AND DE- TERMINATIONS. - ^{2 . 1} .1 The natural line shape is a

(1) We thank Dr. R. Manne for sending us the preprint.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971441

ON THE SHAPE AND WIDTH 0 IF THE MAIN LINES OF X-RAY C4-219

dispersion shape (Lorentz function) [13]. This has been confirmed many times by experiment (see, for example, [14]). Own investigations yielded the following [15] : Allison's overlapping factor f was calculated in dependence on the width and doublet distance for two doublet components (intensity ratio 2 : 1) having a Lorentz and Gaussian shape, resp. It was compared with some 100 measurements of the KP,,, doublets of molybdenum. Since the height of the minimum between both the doublet components going into f, is determined by superposition of the tails of both lines, f is with regard to the respective line shape rather sensitive (Fig. 1). It was evident that the f-values measured were close to the values calculated

w, and w, (the short and long-wave part of the width, respectively), the right and the left half of the line may be described by a half dispersion curve each with the halfwidth w, and w,, resp.

w,

+

.

b) The line shows additional structural details such as splittings, dips, shoulders and others. These details may not be described analytically or by a few para- meters.

2.2 ANALYSIS AND SYSTEMATIZATION OF THE EXPE- RIMENTAL RESULTS. - 2.2.1 The width of the Kci, .,

lines as a function of the ordinal number Z show, as known, in the range of the 3 d elements a considerable deviation with regard to a line broadening. In a diagram we quote the line widths of those authors who measured simultaneously in a larger Z domain [17, 16, 14, 18, 191 ; for a better understanding they are referred to the energy of the corresponding line : AE/E = f ( 2 ) (Fig. 2). There are distinct deviations in the range of the 3 d elements. Further deviations

FIG. 2. - Relative Kal line width in dependence on Z.

FIG. 1. -Doublet superposition for Lorentz and Gaussian shape.

for a dispersion shape, in general, however, somewhat below ; i. e. the tails decline somewhat steeper than with the dispersion line shape.

Deviations of significance from this natural line shape ought to be always due to the overlapping of several dispersion lines (regardless of any instrumental distortions).

2.1.2 Under optimum experimental conditions the instrumental distortions of the lines are negligibly small while measuring with focussing spectrographs of the Johann or Cauchois type. This has been often proved by good agreement of widths and indices of asymmetry with (corrected) values gained from a double crystal spectrometer (e. g. [16, 141 and own measurements).

2.1 .3 Investigating the shape of the main emission lines we distinguish two cases :

a) The line keeps its smooth shape. For the charac- terization it suffices to indicate the ⁽⁽ gross )>parameters

of this nature, however, are not to be discovered (also not for 4 d elements) ; it may be that the measurements in their present state are still too inexact to enable more precise statements. This is true also for the Ka,., doublet distance [20] the changes of which are in causal relation with the changes of the line shape.

For KP,., we know only the early work of Ingel- stam [21] dealing with systematical measurements of width, asymmetry and doublet distance on a wider 2-range, where he did not notice any asymmetry and shows a smooth curve AE = f (2).

The asymmetry indices of the Ka,., lines mentioned by several authors diverge for the single elements up to two units in the first decimal (e. g. [17, 141 and the works quoted in [I] as well as further studies carried out in Leipzig [22,7,23]). The slope in dependence on Z , however, is on the whole uniform.

For 3 d transition elements the indices of asymmetry of the Ka, lines lie considerably above 1 and attain maximum values of 1,40 up to 1,60 (for Mn and Fe) ; for smaller ordinal numbers (2 < 22) the data given are dubious due to the doublet superposition, for larger ordinal numbers ( 2 > 30) there are only a few

04-220 J. FINSTER, G . LEONHARDT AND A. MEISEL indications, in general they are close to the value 1.00

for the symmetrical line (Fig. 3a). Gokhale [16] did not find any asymmetry, Nelson et al. 1191 do not assume any at all. Two different series of measurements in our laboratory yielded for the Mo Kal line values of 1.006 and 0.983. resp. 115,241 and for Ru Ka, 1.02 1;15].

With Ka, lines the asymmetry indices rise only later (recently, we found for Cr Ka, values of 0.97 and 0.92 [7,23]), they do not attain such highvalues and decrease only gradually (Fig. 3 b).

FIG. 3. - Index of asymmetry in dependence on Z a) Kal. b) KEZ

.

2.2.2 Changes of the shape and width of Ka, ,, lines of an element having diferent binding states are again large in particular with 3 d elements, sometimes even larger than line shifts, and they were referred to rather often for our determinations of the binding state. This is exemplified in figure 4 showing a recent measure-

FIG. 4.

-

ment of the Cr Ka,., doublet that was performed in connection with investigations of carrier catalysts [23].

The parameters used by us are indicated in figure 5.

This kind of demonstration (the so-called line cross section) is a good tool for an analysis of the changes of the line shape, especially with regard to a non- resolved multiplet structure.

y + w , = w ; 2 = a

Ws

A y =6,-4; Aw,=&-b;

Aw =A& + A w l = - 4

l all quantities in eVJ

i . ^Aa

w w , b " F

FIG. 5.

-

As known, there is a qualitative correlation between the changes of widths and asymmetry indices as well of the doublet distance and the number u of unpaired (d-) electrons ; so far, there is, however, no quantitative relation. As an attempt in this direction we list the following table presenting measurements of Ka, of the 3 d elements [I] which were carried out in Leipzig on the base of an almost uniform procedure : All broa- dening~, changes of the asymmetry index and of the doublet distance were referred to the compounds with u = 0 and divided by the respective number of unpaired electrons. The relative values i, and in yield a less good picture than the absolute ones Aw and h a and have therefore been omitted. As to the broadening there is on the whole a uniform increase of approxi- mately 0.30 eV per unpaired electron (exempt Mn), the increase of the asymmetry index per unpaired electron diminishes somewhat from Ti to Ni and is in the average about 0.10 ^(2).

Apart from 3 d elements there exist only a few indications as to the changes of shape and width of the main emission lines in dependence on the binding state ([17] for S, [25] for Al, [26] for P, 1271 for Zn, Ga, Ge), the changes are small and allow no generaliza- tions this time.

However, of particular interest is the fact whether the effects occuring with 3 d elsments reappear with 4 d elements, although somewhat lessened. Having stated already before the influence of the chemical bond on the Mo K/3 doublet [15], we investigated now also the Mo Kg,., doublet with increased accuracy. The experimental details will be published elsewhere 1241, a graph is to be seen in figure 4. Table I1 shows nume- rical values. Since the effects for the KP,., doublet of the 4 d elements in energy units have almost the same order as for Ka,., of the 3 d elements, they amount for Ka,., of the 4 d elements only to one fifth (e. g.

for the transition from MOO, to MOO, the line width increases by 0.12 eV, u by 2, i. e. Awlu = 0.06 versus

x 0.30 eV for Ka, of the 3 d elements).

(2) The assignment of the values of the metals to these series enables by the way a rough evaluation of the number of unpaired electrons in the metal ; one obtains :

ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY

Mean values of the changements of line widths, indices OJ asymetry and doublet distances in dependance on the number u of unpaired electrons

Formal Element Valency (*)

- -

Ti 4+

3 + 2+

metal

v

4+

3+

metal 6+

3 + 2+

metal 7 + 6+, (2+)

(3

+)

4+

3+

2+

metal (2+) (3+) 6 + 2 + 3+

metal

Co (2+, 3+)

(2+) 2+

metal

Ni (2+)

2+

metal

(*) In parenthesis : complex compounds.

TABLE I1

Inf[unce of the chemical bond on the M o Kal.2 doublet

(*) In parenthesis : number of measurements.

Ka 1 Kaz

- -

Metallic Mo w (ev> 7.01 7.13

o[ 0.983 0.979

2.2.3 In recent years, several reports were given on structural details of Kcllez lines [27, 1, 28, 29, 30, 311.

Those deep splittings as demonstrated by the first three works on Cr20, and MnF,, resp., appear from the first to be rather unlikely, since the present resolving power does not allow such shapes. Our last measu- rements on Cr yielded a very good resolving power

-

Ka, width in eV : [7] 2.08, [23] 2.01 (cf. Shah, Das Gupta [30] 1.90 ; Schnopper, Kalata 1311 2.09)

-

for Cr2+ and Cr3+ we found only near the maximum a pronounced shoulder (no splitting) - similar to the structure designated by Shah, Das Gupta as (1) and (2) -but we cannot confirm the shoulders denoted as (3) and (4).

3. Theoretical treatment. - 3.1 GENERALREMARKS.

-

All results obtained so far may be regarded as a

(2-222 J. FINSTER, G . LEONHARDT AND A. MEISEL confirmation that deviations of the width of the

main emission lines from the simple sum of the level widths, deviations of their shape from the symmetrical dispersion shape as well phenomena such as the KBl p' structure are chiefly due to a multiplet splitting, i. e. to the splitting of the X-ray terms by interaction of unpaired electrons of non-closed shells. A number of further effects assumed does not play any or only a second-order role (see a consideration published lately by Ekstig et al. [12]).

Although the well based theory of atomic spectra represents the essential fundaments, there is still no sufficently exact calculation of the multiplet structure together with a quantitative interpretation of the measurements. All calculations made up t o now furnish most different approximations. Therefore, they offer only a bad comparability. It is especially difficult to draw any conclusions as to significance and effect of one or the other neglect.

In what follows the attempt shall be made to bring about a certain order into the different stages of approximations beginning with the most exact solu- tion ; the single steps, however, are not always succes- sive.

A. Complete ab-initio calculation for molecules and solids, resp. ; difference of the states 1s dx sY pz

-

2 p5 dX sY pZ ; consideration of relaxation effects (VEC states) and oscillation states.

B. Free ion in adiabatic approximation with or without correlation energy (e. g. by C I).

C. Free ion in sudden approximation with or without correlation energy.

D. Neglect of the s and p fraction of the valence electrons ; neglect of the splitting of the 1 s term.

E. 1. Intermediate coupling or

2. Extreme coupling (LS and jj-coupling, resp.).

F. << Restricted ⁾⁾ coupling (for example, L = 0, i. e. spin-only model or

5,,

The calculation ought to supply as a result respec- tively the prediction and interpretation of structural details provided a sufficient resolving power as well the correct representation of relations with regard to the changes of the gross parameters width and asymmetry found empirically (e. g. for an element in states with ,different numbers of unpaired electrons).

3.2 CLASSIFICATION AND ANALYSIS OF PREVIOUS CALCULATIONS.

-

those of Nefedov 1111 ; they involve investigations of all 3 d elements.

Tsutsumi [8] : Stage C (hydrogen-like functions)

+

+

Horak [9] : Stage C (wave functions after Slater and Watson [32])

+

Israileva [lo] : Stage C (Slater and Hartree functions)

+

+

Nefedov [ l l ] : Stage C (wave functions after Wat- son)

+

+

Ekstig et al. [12] : Stage C (wave functions after Clementi [33])

+

All five works belong to the neglect of stage D (Ekstig [12] gives for the 1 s splitting a value of merely 0.04 eV) ; the correlation energy is taken into account only by Ekstig, but, indeed, it fits less well the experiment (obviously, different approximations com- pensate partly on the stage without CI, as it occurs often with approximations).

It is the merit of Nefedov having treated in detail the intermediate coupling derived from both the extreme cases (17-coupling for KN, ., and LS-coupling for KP, j') and having made the attempt to interpret the changes of shape and width of the Ka, ., lines in depen- dence on the chemical bond. He tried also to consider the influences of solid and molecular states, resp., by cc quenching >> of the angular momentum (L = 0), by introduction of a MO parameter a2 and an effective charge of the ion ; however, it is rather difficult to check the efficiency and quality of these approxima- tions, since all <c theoretical)> values indicated are basing on parameters a of the electrostatic interaction, which were adapted to the experimental results.

The most recent work in this domain (Ekstig et al.

[12]) show that already the calculation on the base of a pure but complete LS-coupling together with <<fro-

zen ^>) wave functions for KP, p' of the 3 d elements

may give rather good agreement with experiment.

3.3 OWN RESULTS. - The inference of the ana- lysis of previous work is that preference should be given to a complete calculation in the framework of a certain stage of approximation (be it a rough one) over the simultaneous, but incomplete application of different approximations ; thus one might gain evi- dence as to the nature and order of the effects and the quality of the approximations.

Our preliminary calculations of the multiplet structure in the approximation of free ions by help of an undisturbed, but complete coupling have been entirely confirmed (3) by the work [I21 for the case of

(3) We used the integrals of Watson [32], Ekstig et a[., those of Clementi [33], the splitting is widely identical.

ON THE SHAPE AND WIDTH OF THE MAIN LINES OF X-RAY LS-coupling (relevant to KP, P' of the 3 d elements)

such that we may confine to considerations of the jj-coupling (relevant to Kotl., of the 3 d and 4 d ele-

ments as well to KP,

.,

3.3.1 General picture of the splitting. - In the simplest case there are 4 components and that two each for LIII 2 Ka, (24 and 16 fold degenerated) and for L,, A Ka2 (12 and 8 fold degenerated).

3.3.2 Values of

c3,

-

p

3.3.3 Splitting of the configuration p5 d.-One obtains 4 terms having the following relative energies [35] :

Term Relative energy (2 J

+

- - -

(3 3

2 2 1,2,3,4 - 4 5,

+

This yields, for example, for Ti3+ the components represented in figure 6. From this splitting results an increase of the width versus the do state (Ti4+) of about 0,08 eV (experiment :

+

there is an increase of the Ka, width versus Mo6+

of x 0.48 eV (experiment only for Mo6+ -+ Mo4+ :

+

+

for the first d-electron, which is compared with the experimental results, too low for the 3 d elements and too high for the 4 d elements.

For the asymmetry the splitting of the term yields for Kol, and KK, equally an index a < 1 ; this is not in good agreement with experiment at least for KK,.

3.3.4 Additional consideration of the electrostatic interaction (intermediate coupling) with the configura-

FIG. 6.

-

tion p5 d. - The insertion of the electrostatic interaction as perturbation yields 8 terms for L,,, and 4 terms for L,,. According to [35] the relative energies of these 12 terms were calculated, for such as Ti3+

(F- and G-integrals from [32]) see table 111.

The picture of the splitting is illustrated in figure 6.

From this we may conclude that width and asymme- try of the integral curve are better compatible with the experimental values of Ka,

.,

Relative energies of the 2 p5 multiplet terms for Ti3+

(jj-coupling with weak electrostatic interaction).

Term Statistical Relative weight energy

- - -

(1 5)

2 2 3 7 - 3.74

(' _{2 2 2} 5) 5 - 3.78

LII (1 3.)

2 2 2 5 - 3.70

(1- 3

2 z ) ~ 3 - 3.60

( 3 a)

+

(4) Having finished this manuscript we got the report from

C. F. Fisher [36] and found the i-values calculated to be of ( ^{2 Z ) I}

) {

about one half of ours. (3. 2 T)O 3 1

+

C4-224 J. FINSTER, G . LEONHARDT AND A. MEISEL 4. Final remarks.

-

of the multiplet structure on X-ray spectra proves to be in general the right and a t present the most success- ful procedure for the interpretation of pecularities of shape and width of the main emission lines ; however, the application was not systematic enough in parti- cular with regard to the introduction of different approximation stages. While the interpretation of line shifts requires absolutely the consideration of atomic surroundings, the calculation of the multiplet structure needs so far only the model of the free ion [12], since

the problem of the correct coupling of the moments is still prepondering.

The complex of the width and transition probabili- ties for the single multiplet components should incite the further discussion.

As to the experiments an improved measuring accuracy is desired as well a better insight into instru- mental influences of the line asymmetry. Since next we may not expect an essential increase of the resol- ving power, the theory ought to give an answer relative to the gross parameters of the line shape.

References [I] MEISEL (A.), Rontgenspektren und chemische Bindung,

Leipzig 1966, p. 212.

[Z] SWAYEV (0. I.), Zh. eksp. teor. Jiz., 1969,57,1716.

Phys. Letters, 1969,30a, 129.

[3] BARINSKI (R. L.), NEPEDOV (V. I.), Rontgenspektros- kopische Bestimmung der Atomladungen in Mole- kulen, German translation, Leipzig, 1969.

[4] LEONHARDT (G.), MEISEL (A.), J. Chem. Phys., 1970, 52,6189.

[5] B L O ~ (M. A.), SHWAYEV (A. T.), Izvest. Akad.

Nauk SSSR, Ser. fiz., 1962, 26,429.

[6] NEFEDOV (V. I.), Zh. Strukt. Khim., 1966, 7, 549 and 719.

[7] LEONHARDT (G.), Thesis, Leipzig, 1969.

LEONHARDT (G.), MEISEL (A.), Spectrochim. Acta, 1970, B 25,163.

[8] TSUTSUMI (K.), ^J. Phys. Soc. Japan, 1959, 14, 1696.

[9] HORAK (S.), Czechoslov. J. Physics, 1960, 10, 405.

[lo] IZRAILEVA (L. K.), Izv. Akad. Nauk SSSR, Ser. fiz., 1961,25,954.

[Ill NEFEDOV ^(V. I.), IZV. Akad. Nauk SSSR, Ser. fiz., 1964, 28, 816 ; Zh. Strukt. Khim., 1964, 5, 650 ; Ibid., 1966,7,719. See also [3].

[12] EKSTIG (B.), KXLLNE (E.), NORELAND (E.), MANNE (R.), UUIP-702, Uppsala, April 1970.

1131 BLOCHIN (M. A.), Physik der Rontgenstrahlen, Berlin, 1957.

[14] MEISEL (A.), NEFEDOW (W.), Z. Chemie (GDR), 1961, 1,337.

[I51 FINSTER (J.), Thesis. Leipzig 1967.

FINSTER (J.), MEISEL (A.), X-ray spectra and electronic structur of matter, Kiev 1969, vol. 11, p. 350.

1161 GOKHALE (B. G.), Ann. de Phys., 1952,7,852.

[I71 PARRATT (L. G.), Phys. Rev., 1936, 50, 1.

1181 BROGREN (G.), Arkiv fys., 1963, 23, 219.

1191 NELSON (G. C.), JOHN (W.), SAUNDERS (B. G.), Phys.

Rev., 1969, 187, 1. UCRL-71856 Add 1.

1201 MEISEL (A.), NEFEDOW (W.), Ann. Physik, 1961,9, 53.

[21] INGELSTAM (E.), Nova Acta Reg. Soc. Sci. Upsal., 1937, IV/10/5.

[22] MEISEL (A.), SZARGAN (A.), Z. physikal. Chemie, 1968, 238,136.

[23] FINSTER (J.), PAUL (R.), MEISEL (A.), to be published.

[24] FINSTER (J.), MEISEL (A.), to be published.

[25] LAUGER (K.), Thesis, Miinchen 1968.

[26] FICHTER (M.), Thesis, Miinchen 1967.

[27] MEISEL (A.), NEFEDOW (W.), 2. Physikal. Chemie, 1962,219,194.

[281 NIGAVEKAR (A. S.), BERGWALL (S.), ^J. Phys. B, 1969, 2,507.

NIGAVEKAR (A. S.), BERGWALL (S.), OHLIN (P.), X-ray spectra

...,

[29] MEISEL (A.), SOMMER (H.), X-ray spectra

...,

[30] SHAH (M.), DAS GUPTA (K.), Phys. Letters, 1969, 29A, 570.

[31] SCHNOPPER (H. W.), KALATA (K.), Appl. Phys. Letters, 1969,15,134.

[32] WATSON-(R: E.), Technical Report No. 12, MIT, 1959.

1331 CLEMENTI (E.), Tables of Atomic Functions, Suppl.

to IBM, J. Res. Develop., 1965, No. 1.

[34] BEARDEN (J. A.), BURR (A. F.), Atomic Energy Levels Oak Ridge, 1965.

[35] CONDON (E. U.), SHORTLEY (G. H.), The Theory of Atomic Spectra, Cambridge 1953.

[36] FISCHER (C. F.), Some Hartree-Fock Results for the Atoms Helium to Radon, QTG, Waterloo

1968.