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Magnetic interactions in heavy atoms
Volume 28A, number
4
PHYSICS
MAGNETIC
LETTERS
INTERACTIONS
2 December
IN HEAVY
ATOMS
1968
*
Z. B. GOLDSCHMIDT and A. PASTERNAK Department
of Theoretical
Physics
and Department
of Experimental
Z. H. GOLDSCHMIDT Physics, the Hebrew University I Jerusalem, Received
4 November
Israel
1968
The inclusion of magnetic interactions in energy level calculations of heavy atoms greatly improved the fit between observed and calculated multiplet splittings. Reliable and consistent values of the appropriate parameters were determined which agree remarkably well with available theoretical predictions.
A systematic investigation was conducted on the effects of magnetic interactions on the energy level schemes of the 3dv configurations (N = = 2,3,. . . ,8) in the third spectra of the iron group, and of PrIV 4f2, PrIII 4f3 and ErIV 4f11. The magnetic interactions considered were the spinother-orbit, the spin-spin and the so called electrostatically correlated spin-orbit [l-3] (effective EL-SO) interactions. Their algebraic matrices were constructed and added to the already existing energy matrices comprising the electrostatic and spin orbit interactions as well as two and three-body effective electrostatic interactions [3-51. The appropriate radial integrals were considered as adjustable parameters, and evaluated by means of the1diagonalization-least-squares * This work was partially supported by the National Bureau of Standards,
Washington,
Magnetic Parameter
D.C.,
interaction
TiIII 3d2
USA.
parameters
procedure. The parameters representing the spin-other-orbit and the spin-spin interactions are the h&s defined by Marvin [6], whereas the effective EL-SO interaction is represented by @s defined as follows: Rk(nltz2,nl?z~l) Qk= (Z((C(k)11Z)2C AEnn’
C&z?zt)
n’
The introduction of the magnetic interactions greatly improved the fit between calculated and observed multiplet splittings in all investigated configurations, with the exception of ErIV 4flI, which was but slightly effected. Space limitations force us to defer the detailed discussion to later publications. The values obtained for the Mks and the Qks are given in tables 1 and 2, for the iron group and the rare-earths respectively. The tables
Table 1. for 3dNconfigurations
in the iron group (cm-i)
VIII 3d3
CrIII 3d4
MnIII 3d5
Fe111 3d6
Co111 3d7
NiIII 3d8 644
c3d
129
177
239
328
411
520
c3dHF
126
184
258
333
426
539
0.498
0.816
1.452
1.770
2.088
2.406
MO HF
0.704
0.915
1.156
1.433
1.704
2.022
2.375
MSS = MZOO
0.229
0.422
0.615
0.808
1.001
1.194
1.387
M2 HF
0.384
0.499
0.631
0.783
0.930
1.103
1.295
Q2,Q4
-1.2
12.7
26.6
40.5
54.4
68.3
82.2
ML
= vi00
1.134
672
265
Volume
Magnetic
28A,
PHYSICS
number 4
PI-1114f3*
PrIV 4f2
ErIV 4fll 2271
C4f
691
788
<4f HF
838
878
2610
M&
1.96
2.14
5.58
MO HF
2.08
2.24
4.96
4s =400
1.21
1.37
3.13
M2 HF
1.17
1.26
2.78
4s =400
0.23
0.75
2.13
0.79
0.85
1.89
68.6
50.8
503.8
= M:oo
M4 HF = Q4
= Q6
* HF parameters of the equality
for &III
4f3 were derived
by means
1968
P(PrII14f3, _ P(EuII14f7) P(NdIV 4f3) - P(GdIV 4f7i
clearly demonstrated the internal consistency of the results obtained, which also are in complete agreement with all available theoretical predictions. The following conclusions concerning the magnetic interaction parameters may be drawn: a) They are all positive, the Mks decreasing as functions of k, whereas the Qks seem to be approximately independent of k (For this reason, the Qks, in each spectrum, were held equal in the final stages of the calculations). b) Their values increase as functions of the atomic number at a rate which agrees pith their theoletical Z dependeice, i.e. Mk m Z,ff, Qk m Thus while the relative cc ‘e f whereas C m Zeff. effec f of the @s diminishes on raising the atomic
**
266
2 December
number, the Qks gain in importance. c) A comparison of the results for PrIII and PrIV shows that the Mks, like most internal parameters, increase with the degree of ionization, whereas the Qks decrease, in common with effective electrostatic parameters. d) The values obtained for the Mks by our methods are in excellent agreement with the results of Hartree-Fock calculations by Blume et al. [?‘I. To the best of our knowledge this is the first example in heavy atoms of such a close agreement between semi-empirical parameters and those calculated from first principles. It must be emphasized that the above results crucially depend on the simultaneous introduction of all three magnetic interactions. The omission of any of these interactions leads to wrong values for the remaining parameters.
Table 2. interaction parameters for the 4fN configurations in the rare-earths (cm-l)
Parameter
Q2
LETTERS
References 1. B. R. Judd. H. M. Crosswhite and Hannah Crossahite, Phys. Rev. 169 (1968) 130. 2. J. Stein, Ph.D. Thesis, The Hebrew University. Jerusalem. Israel (1967). A. Pasternak and Z. H. Goldschmidt 3. 2. B. Goldschmidt, unpublished material. 4. Y. Shadmi. E. Caspi and J. Oreg. private communication. Phys. Rev. 141 (1966) 4. 5. B.R.Judd. Phys. Rev. 71 (1947) 102. 6. H.H.Marvin, 7. M. Blume and R. E. Watson. Proc. Roy. Sot. London A271 (1963) 565; M. Blume. A. J. Freeman and R. E. Watson. Phys. Rev. 134 (1964) A320.
***
4
PHYSICS
MAGNETIC
LETTERS
INTERACTIONS
2 December
IN HEAVY
ATOMS
1968
*
Z. B. GOLDSCHMIDT and A. PASTERNAK Department
of Theoretical
Physics
and Department
of Experimental
Z. H. GOLDSCHMIDT Physics, the Hebrew University I Jerusalem, Received
4 November
Israel
1968
The inclusion of magnetic interactions in energy level calculations of heavy atoms greatly improved the fit between observed and calculated multiplet splittings. Reliable and consistent values of the appropriate parameters were determined which agree remarkably well with available theoretical predictions.
A systematic investigation was conducted on the effects of magnetic interactions on the energy level schemes of the 3dv configurations (N = = 2,3,. . . ,8) in the third spectra of the iron group, and of PrIV 4f2, PrIII 4f3 and ErIV 4f11. The magnetic interactions considered were the spinother-orbit, the spin-spin and the so called electrostatically correlated spin-orbit [l-3] (effective EL-SO) interactions. Their algebraic matrices were constructed and added to the already existing energy matrices comprising the electrostatic and spin orbit interactions as well as two and three-body effective electrostatic interactions [3-51. The appropriate radial integrals were considered as adjustable parameters, and evaluated by means of the1diagonalization-least-squares * This work was partially supported by the National Bureau of Standards,
Washington,
Magnetic Parameter
D.C.,
interaction
TiIII 3d2
USA.
parameters
procedure. The parameters representing the spin-other-orbit and the spin-spin interactions are the h&s defined by Marvin [6], whereas the effective EL-SO interaction is represented by @s defined as follows: Rk(nltz2,nl?z~l) Qk= (Z((C(k)11Z)2C AEnn’
C&z?zt)
n’
The introduction of the magnetic interactions greatly improved the fit between calculated and observed multiplet splittings in all investigated configurations, with the exception of ErIV 4flI, which was but slightly effected. Space limitations force us to defer the detailed discussion to later publications. The values obtained for the Mks and the Qks are given in tables 1 and 2, for the iron group and the rare-earths respectively. The tables
Table 1. for 3dNconfigurations
in the iron group (cm-i)
VIII 3d3
CrIII 3d4
MnIII 3d5
Fe111 3d6
Co111 3d7
NiIII 3d8 644
c3d
129
177
239
328
411
520
c3dHF
126
184
258
333
426
539
0.498
0.816
1.452
1.770
2.088
2.406
MO HF
0.704
0.915
1.156
1.433
1.704
2.022
2.375
MSS = MZOO
0.229
0.422
0.615
0.808
1.001
1.194
1.387
M2 HF
0.384
0.499
0.631
0.783
0.930
1.103
1.295
Q2,Q4
-1.2
12.7
26.6
40.5
54.4
68.3
82.2
ML
= vi00
1.134
672
265
Volume
Magnetic
28A,
PHYSICS
number 4
PI-1114f3*
PrIV 4f2
ErIV 4fll 2271
C4f
691
788
<4f HF
838
878
2610
M&
1.96
2.14
5.58
MO HF
2.08
2.24
4.96
4s =400
1.21
1.37
3.13
M2 HF
1.17
1.26
2.78
4s =400
0.23
0.75
2.13
0.79
0.85
1.89
68.6
50.8
503.8
= M:oo
M4 HF = Q4
= Q6
* HF parameters of the equality
for &III
4f3 were derived
by means
1968
P(PrII14f3, _ P(EuII14f7) P(NdIV 4f3) - P(GdIV 4f7i
clearly demonstrated the internal consistency of the results obtained, which also are in complete agreement with all available theoretical predictions. The following conclusions concerning the magnetic interaction parameters may be drawn: a) They are all positive, the Mks decreasing as functions of k, whereas the Qks seem to be approximately independent of k (For this reason, the Qks, in each spectrum, were held equal in the final stages of the calculations). b) Their values increase as functions of the atomic number at a rate which agrees pith their theoletical Z dependeice, i.e. Mk m Z,ff, Qk m Thus while the relative cc ‘e f whereas C m Zeff. effec f of the @s diminishes on raising the atomic
**
266
2 December
number, the Qks gain in importance. c) A comparison of the results for PrIII and PrIV shows that the Mks, like most internal parameters, increase with the degree of ionization, whereas the Qks decrease, in common with effective electrostatic parameters. d) The values obtained for the Mks by our methods are in excellent agreement with the results of Hartree-Fock calculations by Blume et al. [?‘I. To the best of our knowledge this is the first example in heavy atoms of such a close agreement between semi-empirical parameters and those calculated from first principles. It must be emphasized that the above results crucially depend on the simultaneous introduction of all three magnetic interactions. The omission of any of these interactions leads to wrong values for the remaining parameters.
Table 2. interaction parameters for the 4fN configurations in the rare-earths (cm-l)
Parameter
Q2
LETTERS
References 1. B. R. Judd. H. M. Crosswhite and Hannah Crossahite, Phys. Rev. 169 (1968) 130. 2. J. Stein, Ph.D. Thesis, The Hebrew University. Jerusalem. Israel (1967). A. Pasternak and Z. H. Goldschmidt 3. 2. B. Goldschmidt, unpublished material. 4. Y. Shadmi. E. Caspi and J. Oreg. private communication. Phys. Rev. 141 (1966) 4. 5. B.R.Judd. Phys. Rev. 71 (1947) 102. 6. H.H.Marvin, 7. M. Blume and R. E. Watson. Proc. Roy. Sot. London A271 (1963) 565; M. Blume. A. J. Freeman and R. E. Watson. Phys. Rev. 134 (1964) A320.
***