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MAGNETIC AND EFFECTIVE INTERACTIONS IN IN CONFIGURATIONS OF HEAVY ATOMS
Z. Goldschmidt
To cite this version:
Z. Goldschmidt. MAGNETIC AND EFFECTIVE INTERACTIONS IN IN CONFIGURA- TIONS OF HEAVY ATOMS. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-163-C4-166.
�10.1051/jphyscol:1970427�. �jpa-00213882�
ETUDES EMPIRIQUES - PROBABIL I TES DE TRANS1 TION
MAGNETIC AND EFFECTIVE INTERACTIONS IN IN CONFIGURATlONS OF HEAVY ATOMS (*)
Z. B. G O L D S C H M I D T
D e p a r t m e n t o f Theoretical Physics T h e Hebrew University, Jerusalem, Israel
Resume. - Une kt~rde systcmatiqi~c a Pte faite concernant les effets dcs interactions ~nagnetiques sur les schenias dc n i v e a ~ ~ x d'knergie des config~~rations 3 d \ (A1 - 2. 3, .... 8) dans les 3 e et 4 e spec- tres du groupe du fer. Les interactions magnktiq~~es considi.rees Ctaient. les interactions spin-autre- orbite, spin-spin ct les efl'ets corrklks du couplagc spin-orbitc ct de I'interuction coulombienne (Effective EL-SO). Lcs intkgralcs radialcs ;ipproprikes .%It (k - 0. 2) et Q" (k = 2, 4) etaient consi- derees comme des paramctres njustablcs ct cvaluccs par des metliodcs dc dii~gonalisation et de nioindrcs carres.
L'introduction des interi~ctions ~ i i i ~ g n c t i c l ~ ~ c s ;I gri~ndcment anieliorc I'accord cntre les decompo- sitions de m ~ ~ l t i p l c ~ s c;ilculi..; ct ubse~.vCs clan\ tu~rtch Ic\ configurations i.r~~~liCcs.
Des vale~lrs pla~~.;ihlcs ct colii.rcntcs cles ~ x ~ ~ . ; ~ ~ i i C t r c s ;~ppropriCs ont ohtcnues ; elles concor- dent reniarqi~al.lcmcnt ii\cc Ics ~?ri.v~sions tlii.oriclucs dont o n dispose. 1-c mode de variation des divers parametres en function clc X . I ~ L I I . cli'pc~id;~~ice ~.cl;~ti\'e~iic~it :I ct It degre d'ionisation ont cti. dctcr~iiinC.s.
Abstract. - i\ s\stcmatic in\c\rig:itioli \ \ a s concluctcd on the ctkcrs uf m:lgnctic intcrac[~c;ns on the cncrgy level schc~lics of the 3 J ' configurations ( . \ ' - - 2. 3. .... 8 ) in the ~ l i i ~ x l and fourth spectra of the iron group. The magnetic interactions conidcrcd \\ere the spin-other-orbit, the spin-spin and the so called clcctrost:itic;illy co~.rclateJ \pin-orbit (cll'eclive EL-SO) i~iteractions. The appro- priate radial integrals ?\I1, ( X -- 0. 2) and Q L ( A 2 . 4 ) \\ere considered as adjustable parameters, and e v a l ~ ~ a t e d b) means of the di:~gonali~ation-Icast-sq~1arc3 p r o c c d ~ ~ r c .
Tlie introd~rction of tlie magnetic inter-iictions greatly impro\,cd the fit bct\\ccn calculated and observed m ~ ~ l t i p l c t splittings in all investigated config~~rations.
Reliable and consistent \.slues of the appropriate pur;imctcrs \\ere obtained. which agree reniar- kably well with available tlicorctic:~l prediction;, 7'hc ~ i i o d c of change of the \,arious parameters with k, and their dependence on Z , . : , a n d tlic degree of ioni7ation \\,ere determined.
U p t o recent years. energy-lc\cl cnlculntions of' heavy a t o m s were i ~ s u a l l y bused on a Hamiltonian which comprised only tlic c o u l o m b intcrnction bet\\,ccn the electrons a n d their indi\ i d i ~ a l spin-o~.bit interac- tions. T h i s led t o a ratlicr p o o r ;lgrccmc~it bet\vcen calculated ancl obscr\.cd lei.,-Is. c \ c n \\lien all pertinent parameters \\,ere considcrcd a s I'rccly :idjustable.
Substantinlly bcttcr lits I'oi- I \' c v n t i g ~ ~ r a t i o n s were obtained. t h r o u g h t h e inclusion in their cncrgy matrices 01' L \ V O ; ~ n d thl-cc-clccr ran clectrostntic effective interactions [ I . 2. 3. 4. 51. T h e mean e r r o r thereby diminished t o se\.crnl tens o f c n i - I . Tlie remaining deviations often were o f a p r o n o u n c e d magnetic c h a r a c t e r , t h u s suggesting t h e nced f o r inclusion in tlie H a m i l t o n i a n o f tlie still missing m u t u a l magnetic interactions. T h e s e intcractions a r e p r o v i d e d by t h e Breit e q u a t i o n . which, in tlie non-relativistic limit, breaks L I P i n t o p a r t s which car1 be given a
siinplc physical interpretation. S o m e o f these. sucli as t h e orbit-orbit nncl t h e spin-spin contact interac- tions. a r e a b s o r b e d by tlie parameters rcprcscnting t h e ulicady included intcractions. T h e parts which :tre not ahsoih:d a r c tlie spin-spin a n d tlic spin-otlier- orbit interactions, which respectively represent t h e
~ n u t u a l interaction bctwccn tlie magnetic dipole m o m e n t s 01' tlic elcctl-ons a n d between t h e dipole
~ n o m e n t 01' o n e elcc:ro~i arid tlie orbital m o t i o n of' a n o t h c r .
T h e o p e r a t o r s representing t h e spin-spin arid tlic spin-ot her-orbit interactions f o r I configurations a r e given by t h e following f o r m u l a e :
I., I
(*) This w o r k \\as partially supportcd by the National + . I + 3 ( [ c ) ~ ) x c . ~ ~ + ~ ) ] ' ~ ' . [ ~ , x ,sj]f2))
Bureau of Standards. Washington D. C.. U. S . A. v
1
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970427
with
The appropriate radial integrals are those defined by Marvin ( 6 )
For d" configurations k takes on the two values 0 and 2 whereas for fh' configurations k = 0, 2, 4.
I n view of the importance of the cfl'ectivc clectro- static interactions as compared to the real ones, the question now arises whether any effective magnetic interactio~zs should be introduced in the tlleoretical calculations alongside the real magnetic interactions.
Indeed, on considering the two dominant terms of the Hamiltonian for heavy atoms, which are the elec- trostatic interaction G and the spin-orbit interaction F, it turns out that the effective Hamiltonian is given, to second order of perturbation theory, by the fol- lowing equation
Hcff = - (C + F ) (C; + F)IAE
AE being the energy separation between the perturbing and the perturbed configurations. This Ha~niltonian dcco~nposes into three parts :
/IA = - GGIAE H , = - ( G F + FC;)!AE
H , = - FFIAE
H , represents the effective electrostatic interaction already included in the calculations. H c t u r ~ l s out to have the same angular dependence as the usual spin- orbit interaction [7, 81. The tensor operator form of H , is given below :
-
- - 2 1 Q " [ I ( ~ + 1 ) ( 2 1 + 1)]"(2 k + I)-!'
k even
I odd
where the parameters Qk are defined as follows :
~ ~ ( n l n l , nln' 1) [,(iln1) Q" ((I )I C '" 11 I)' 2 --
n' AE",,, ( 5 )
with R ~ I I / I I ~ , I I / ~ ' I) and [,(nnl) being respectively a Slater and a spin-orbit parameter ; also,
where d" arc the unit tensor operators defined by Racah [9]
a n d
k takes on the two sets of values 0, 2, 4 and 0, 2, 4, 6 for d N and f N configurations respectively, but since the k = 0 term has the same angular dependence as the spin-orbit interaction, i t is dropped from the calcu- lations. From now on we shall reler to the interaction represented by HB as (( Elrectivc electrostatic -- spin- orbit interaction )) (EfTective EL-SO).
A systematic investigation was conducted on the cfl'ects of the spin-spin, spin-other-orbit and efl'ective EL-SO interactions on the energy level structure ol' the 3 d" configurations (IV = 2, 3, ..., 8) in the third and fourth spectra of the iron group [lo, 11, 121. For this purpose the algebraic matrices of these interac- tions werc constructed and added to the already existing energy matrices. The appropriate radial integrals werc considered as adjustable parameters.
and evaluated by means of the diagonalization least- squares procedure. The analysis was conducted in two stages. 111 the first stage each configuration was separately treated, whereas in the secorld stage a general least-squares (GLS) calculation was per- formed, treating all the 3 d f V configurations as a single probelm, in which all the parameters except were constrained to change linearly with N : for ;,,
a quadratic correction term was also includccl. Thc linearity of the iMA s and QA s was deducecl fro111 the results of the separate calculations of the individual spectra. In addition, the I'ollowing restrictions were imposed on the rnagnctic interaction parameters :
These werc deducecl f'ro~n the results 01' preliminary calculations in which all magnetic parameters were permitted to vary independently. the first one also being suggested by the ddinition 01' the M A s.
The introduction ol' the mutual magnetic and effective EL-SO interactions greatly improved the fit between calculated and observed multiplet splittings in all investigated configurations. Typical examples arc shown in figures 1 and 2. As demonstrated by these figures, the effects of the added iilteractions is to practically equalize the deviations between calculated and observed energy levels, for all levels belonging to the same multiplet. Thus deviations of magnetic character are eliminated and the remaining cievi:i~io~~s arc mainly of a purely elcctrost:ltic character. These residual deviations prevent the mean error obtni~lcd
MAGNETIC A N D EFFECTIVE INTERACTIONS IN I-' CONFIGURATIONS O F H E A V Y ATOMS C1-165
in the least-squares calculations from reflecting the improvement in the fit due to the mutual magnetic and effective EL-SO interactions. A new criterion to measure this improvenlent has therefore been intro- duced, which we call (( Observed minus calcula- ted (0-C) spread D. This is defined as the absolute value of the difference between the maximum and minimum deviations for levels belonging to the same multiplet. For the multiplets given in figures 1 and 2, the reductions in the 0 - C spread are given below :
A NO MI INCLUDING MI
On the whole, the sum of 0 - C spreads has reduced from 786 cm-' to 274 cm-' for the third spectra of the iron group and from 806 cm-' to 201 cm-' for the fourth spectra.
The values obtained for the M% and Q% are given in tables I and I1 for the third and fourth spectra of the iron group respectively. The tables clearly demonstrate the internal consistency of the results obtained, which also are in complete agreement with all available theoretical predictions.
FIG. 1. - Improvement in 3 d4 3H multiplet splitting due to mutual magnetic and effective EL-SO interactions.
A NU MI
* INCLUDING MI
0 - c crn-'
The following conclusions concerning these para- meters may be drawn :
a) They are all positive, the M k s decreasing as functions of k, whereas the Q k s seem to be approxi- mately independent of k.
b) Their values increase as functions of the atomic number at a rate which approximately agrees with their theoretical Z dependence, the M k s increasing as z:~, in, as z:, and the Qk s, which comprise sums of products of [,(nnr) and R~ as z&. Thus, while the relative effect of the M k s diminishes on raising the atomic riumber, the ek s gain in importance.
FIG. 2. -Improvement in 3 ds 4G multiplet splitting due to mutual magnetic and effective EL-SO interactions.
Magnetic and egectiue EL-SO parameters for 3 dN configurations in tlte third spectra of the iron group (cm-')
Parameter
-
Z. H. GOLDSCHMIDT
Magnetic attd effecfiue EL-SO paratneters for 3 d N confgurations in tltefourth spectra of the iron group (cm-')
Parameter V IV 3 d 2 C r lV 3 d 3 Mn IV 3 d 4 Fe IV 3 d 5 C o IV 3 d 6 Ni IV 3 d 7 C u IV 3 d R
--- - - - - -- -- -
c ) A comparison of t h e results for t h e third and three interactions. T h e omission of the emective EL-SO
fourth spectra shows that the M h s, like most internal leads t o incorrect values and wrong ratios of thc M A \.
parameters, Increase with the dcgl-ee of ionization, M O being considerably smaller than M ' . Calculatlo~l whereas the QA s decrease, in common with thc efTec- in which EL-SO only was considered yielded complc- tive electrostatic parameters. tcly i~nrcnsonable values for the Q".
(/) The values obtained for the M\ by o u r methods The effects o f t h e mutual magnetic and e f i c t i v e EL- are in excellent agreement with the results of Hartree- S O intcractions o n the energy level structure of 4 1"
Fock calculations by Blume and Watson [13]. configurations in several third and fi)ilrth rare-earth It must be emphasized that the above results cru- spectra were also investigated [ l o , 14. 151. TIlese \till
cially depend on the simultarleous introduction of all be disc~lssed elsewhere.
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