EXCHANGE POLARIZATION OF THE 3 d-SHELL AND ISOTOPE SHIFT OF THE LEVELS Cu II 3 d9 4 s 1D AND 3D

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EXCHANGE POLARIZATION OF THE 3 d-SHELL AND ISOTOPE SHIFT OF THE LEVELS Cu II 3 d9 4

s 1D AND 3D

M. Elbel, H. Hühnermann

To cite this version:

M. Elbel, H. Hühnermann. EXCHANGE POLARIZATION OF THE 3 d-SHELL AND ISOTOPE SHIFT OF THE LEVELS Cu II 3 d9 4 s 1D AND 3D. Journal de Physique Colloques, 1969, 30 (C1), pp.C1-41-C1-47. �10.1051/jphyscol:1969111�. �jpa-00213642�

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Colloque C I , supplPment au no I , Tome 30, Janvier 1969, page C 1

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EXCHANGE POLARIZATION OF THE 3d-SHELL AND ISOTOPE SHIFT OF THE LEVELS Cu IZ 3 d9 4 s ' D AND 3 D

M. ELBEL a n d H. H ~ ~ H N E R M A N N Physikalisches Institut, MarburgILahn, Renthof 5

Rhumb. - D'apres la theorie du lcr ordre de l'cffet de masse specifique, les niveaux ID et 3 0

de la configuration 3 d9 4 s dans le spectre de Cu I1 (et de Ni I ) devraient avoir le mCme deplace- ment isotopique. Des mesures indiquent cependant des differences qui s'elevent a ⁴mK dans le cas du cuivre ( 6 M = 2) ct a 4 mK dans le cas du nickel ( 6 M = 6). Nous avons essaye d'attribuer ce comportement h une difference entre les orbitalcs 3 d dans les etats ID et 3D respectivement.

Dans un modele d'interaction de configuration, cela peut Ctre decrit par des melanges de niveaux 3 d8 4 d 4 s aux niveaux 3 d9 4 s. Les amplitudes de ces melanges different pour les etats ID et 3 0 .

Elles sont proportionnelles au parametre de Slater R2(3 d 4 s, 4 s 4 d ) et inversement proportionnelles A la distance des configurations 3 dY 4 s et 3 d8 4 d 4 s dans I'approximation d'ordre zero.

Ces melanges modifient l'effet de masse specifique de la couche 3 d, avec des couches 2 p et 3 p plei- nes, de quantites qui sont proportionnelles A J(3 d, 2 p). J(4 d, 2 p) et h J(3 d, 3 p ) . J(4 d, 3 p ) oh les symboles J designent les integrales bien connues introduites par Vinti. Ces intkgrales ont kt6 calcul&s numeriquenient en utilisant des fonctions de Hartree-Fock, les unes donnees par Hartree, les autres resultant de nos propres calculs. L'accord entre thcorie et experience est satisfaisant et de nature a klairer d'un jour nouveau I'origine des deplacements isotopiques dans cette region du systeme periodique.

Abstract. - According to the first-order theory of the specific-mass erect the ID- and 3D-levels of 3 d9 4 sconfiguration in the Cu II- (and Ni I-) spectrum are expected to have the same isotopic displacement. Measurements indicate, however, that there is a difference which amounts to 4 mK in the case of copper ( 6 M = 2) and 4 mK in the case of nickel (6M = 6). An attempt is made to ascribe this behaviour to a difference in the 3 d-orbitals of ID and 3D states, respectively.

Within a configurational interaction model this may be decribed by admixtures of 3 d 8 4 d 4 s states to 3 d9 4 s states. The amplitudes of these admixtures differ in the cases of ID and 3D states.

They are proportional to the Stater parameter R2(3 d 4 S, 4 s 4 d ) and inversely proportional to the distance of the configurations 3 d9 4 s and 3 dx 4 d 4 s in the zero-order approximation. These admixtures alter the specific-mass effect of the 3 d-shell with the filled 2 p-and 3 p-shells by amounts which are proportional to J(3 d, 2 p)-J(4 d, 2 p ) and J(3 d, 3 p).J(4 d, 3 p), where the symbols J denote the well-known integrals of Vinti. These integrals are calculated numerically by means of Hartree-Fock functions which have been given partly by Hartree, partly they result from our own calculation. The agreement of the measured and thecalculated values is satisfactory and apt to give a new impression of the origin of isotopic shifts in this region of the periodical system.

1. Introduction. - T h e first-order theory of mass- dependant isotope-shift which has been worked out by Hughes a n d Eckart [I], Bartlett a n d Gibbons [2]

a n d Vinti [3] can explain some features of the measured shifts in the spectra of light elements. But there arc examples where it failes, as well. For instance this theory predicts that all the levels of a n electronic configuration should have the same shift provided the following condition holds : T h e configuration should not contain electrons with orbital quantum numbers differing by exactly one unit. Closed shells are excluded from this condition. In accordance with this rule the levels 4 s 4 p ' P and 4 s 4 p ^3~ in Z n I-

spectrum possess unequal isotope shifts. Conse- quently this phenomenon had been referred t o the existance of a Hughes-Eckart-integral J ( 4 p, 4 s ) by Crawford a n d others [ 4 ] .

Measurements, however, in the Cu 11-spectrum [ 5 ] a n d in the Ni I-spectrum [6] indicate that the levels 3 d9 4 s 'D a n d 3 d9 4 s 3 D possess unequal shifts, a s well. This phenomenon contradicts the quoted rule. T h e results are in detail :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1969111

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for the nuclide Ni64 as compared to the nuclide Ni5*

for the nuclide C U as compared to the nuclide ~ ~ C U ~ ~ . (Positive isotope shift means that the hypothetical transition 3 d9 4 s ' D + 3 d9 4 s 3~ is shifted towards higher frequencies for the heavier nuclide as compared to the lighter one.)

There are other topics which disagree with Hughes' and Eckart's theory. Look at figure 1. There the resi-

FIG. 1. -Residual shifts of some configurations of the Cu I1 spectrum.

dual shift of Cu 11-3d9 nl levels is plotted against the mean distance of the nl electron from the nucleus.

Residual shift means isotope shift less normal shift.

Therefore it equals the specific mass effect plus volume effect. The residual shift increases from 4 s to 4 p.

Then it remains constant to even higher excited electrons thus representing the residual shift of the ionization limit. But the Hughes-Eckart-theory would predict that the residual shift should decrease instead of increasing when the 4 s electron is removed because then it would no longer contribute to the total spe-

m M

cific mass effect with the terms 2 A Ry J2(np 4s).

M1M2 ⁿ

Recent calculations of Bauche [7] concerning specific mass effect in Cu I-spectrum ( I ) show that the rise of the specific mass effect accompanying the excitation of the 4 s-electron is due to changes of the 3 d-eigenfunctions. The radial functions of the 3 d-electrons are affected by the removal of the 4 d-electron in such a way that the Hughes-Eckart integrals J(3 d, 2 p) and J(3 d, 3 p) are increased markedly thus accounting for the increase of the total specific mass effect. Bauche's calculations have been repeated by the authors with Cu I-eigenfunctions of Synek [8]. The results are given in table I exhibiting clearly the behaviour just described. The agreement with Bauche's results is fairly good.

Let us now consider how the 4 s electron influences the 3 d electrons. The strongest influence arises from the direct screening of the 3 d electron by the 4 s charge cloud. But the exchange interaction between 3 d and 4 s electrons should not be overlooked completely because it is the latter kind of interaction which may influence the 3 d electrons of the ' D and the 3D in a different way. This we need in order to explain the different isotope shift of ' D and ³ ^~ ^.

FIG. 2. - Calculated radial functions of some s electrons in the copper ion.

(1) The spzcific mass effect in the C u I-spectrum increases in the same manner as it does in the Cu II spectrum when the 4 s electron is exited or even removed.

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Calculated specific mass effects of some Cu 1 states. Synek's [El functions were used. I2 denotes JZ(n1, n'l - I), nl, n'l - 1 being specified at the top of the table.

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The configuration 3 d8 nd 4 s gives rise to a variety of 'D and 3D states which all may share in the per- turbation of 3 d9 4 s 'D and 3 ~These are .

1. f o r 3 d 9 4 s 1 D

( 3 d 8 ) 1 ~ ( n d 4 s ) 1 ~ 1 ~ (3 d8) 3 ~ ( n d 4 s ) 3 ~ 'D ( 3 d 8 ) 1 ~ ( n d 4 s ) 1 ~ ' ~ ( 3 d 8 ) 3 ~ ( n d 4 ~ ) 3 ~ ' ~ (3 d8) ' ~ ( n d 4 s) 'D

2. for 3 d9 4 s ^3~

I ( 3 d 8 ) 1 ~ ( n d 4 s ) 3 ~ 3 ~

Fig. 3. - Calculated radial functions of some d-electrons

in the copper ion. ( 3 d 8 ) 3 ~ ( n d 4 s ) ' ~ 3 ~ and ( 3 d 8 ) 3 ~ ( n d 4 s ) 3 ~ 3 ~

. . . . ,

2 . Configuration interaction between 3 d9 4 s and

3 d 8 nd 4 s. - The difference between the 3 d-radial ( 3 d 8 ) 3 ~ ( n d 4 s ) ' ~ 3 ~ and ( 3 d 8 ) 3 ~ ( n d 4 s ) 3 ~ 3 ~ wave functions of 3 d9 4 s and 3 d9 4 s 3~ can be (3d8) ' ~ ( n d 4 s ) ^3~ 3D.

easily described by slight admixtures of the states

Generally we shall write 3 d8 nd 4 s ' D and 3 d8 nd 4 s 3 ~ , respectively. n means

any quantum number higher than three. Another (3 d8) SL(nd 4 s) S' L! ID, 3~ .

possibility would consist in unrestricted Hartree-

Fock-calculations. We are choosing the first way which These states occur with amplitudes in the perturbed seems to be less laborious. states ID, 3D (') which can be given to the first order

by the equation

--

< C(3 d8) SL(nd 4 S) Sr I!] 'D i I D > = (3 d 8 ( S ~ ) 3 d 'D } 3 d9 '0) J2(2 Sf + 1) W(S 3 0 9,3 St) x

< [(3 d8) SL(nd 4 s) S' L'J ^3~ 1 3~ > = (3 d 8 ( S ~ ) 3 d 'D } 3 d9 '0) J2(2 Sf f 1) W ( S 4 1 + , 4 S') x

The amplitudes eq. (3) and (4) were calculated by evaluating the matrix elements of the electrostatic interaction of 3 d9 4 s ID, ^3~ with anyone of the excited states of (1) and (2). In order to take advantage of the selection rules of this interaction one decouples the interacting 3 d-electron of 3 d9 4 s 'D, 3D coupling it afterwards to the 4 s-electron. Thus one expands the states (3 d9 4 s) 'D, 3D to the states

(3 d8) SL(3 d 4 s) S' L' 'D, 3~ .

The first coefficient denotes a well-known ^(<coefficient of fractional parentage )) occurring when one 3 d electron is decoupled froin the 3 d9 shell. The second coefficient which is a Racah-coefficient results from the recoupling of the spin of the 3 d electron. The

coefficient f i results from the fact that each of the nine 3 d electrons of 3 d9 4 s may interact with the nd-electron of 3 d8 nd 4 s. This point is more tho- roughly discussed in a paper of Rosenzweig [9].

The RO and R2 are Slater-parameters the first of which accounts for the 3 d-nd excitation by direct screening from the 4 s electron. The second one accounts for the 3 d-nd excitation by exchange polarization of the 3 d shell by the 4 s electron. Concerning the energy denominator it had been assumed that

( 2 ) In the following the perturbed states which have been

classified after 3 d9 4 s ID, 3 0 will be denoted by ID, 3 0 without any specification of coniiguration.

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EXCHANGE POLARIZATION OF THE 3 d-SHELL AND ISOTOPE SHIFT C 1- ⁴⁵

the energetical difference of all the levels of 3 d 8 nd 4s 3. Non diagonal matrix elements of the specific mass with respect to the levels of 3 d 9 4 s roughly equals effect with respect to the codgurations 3 d 9 4 s and the excitation energy of a single 3 d electron to the 3 d 8 nd 4 s. - Non diagonal matrix elements of the nd level. The coefficients of the expansion of specific mass effect exist between 3 d9 4 s and i d 9 4 s and 3 d 8 nd 4 s. They arise if the operator of (3 d 9 4 s ) ID, 3 D to [(3 d 8 ) SL(3 d 4 8) S' L ' I I D , 3D the specific mass effect

have been given in table 11.

TABLE TI

Coefficients which occur when the states (3 d9 4 s ) ID, 3 0

are expanded to the states [(3 ds) SL(3 d 4 s) S' L' = 21 ID, JD.

S S' ~ ( 3 d s ( ~ ~ ) 3 d 2 ~ } 3 d 9 2 D ) , / ~ ~ ( ~ : 0 * ; f ~ ' )

0 0 0 2/1/45

1 1 1 J9/45

0 0 2 J5/45

1 1 3 J21/45

0 0 4 J9/45

N = total number of electrons in the atom puts into correlation an inner p { i ) electron of 3 d 9 4 s with the nd { i }-electron of 3 d 8 nd 4 s on the one hand and an inner p { j ) electron of 3 d 8 nd 4 s with any 3 d{ j ) electron of 3 d 9 4 s on the other hand.

The superscripts { i ) and { j } denote the number of the electron's coordinates. All other electronic states should have the same electronic coordinates in both configurations for otherwise the matrix element of 6(0) vanishes. The detailed calculation of the matrix'element is greatly facilitated by the fact that

" (3 ds(SL) 2D d9 2D) J 2 ( 2 S ' + 1 ) w(S*

*

^;^*") the inner p-electron belongs to a closed shell. For this

0 1 0 J1/45 reason the matrix elemen6

are all degenerate with respect to S L , S' L', S" L"

and equal the same value

& (415) Ry(m, 8 M I M l M 2 ) C - ^.&. J(3 d, mp) J(nd, mp) .

From this one concludes

/ ( 3 ~ Y s L ) ~ d % } 3 d9 ' D ) \/2(2 S1 + 1) W ( S t 0 f, f S') x

x dg (415) Ry(me ,GM/M1 M 2 ) ( ~ ( 3 d , 2 p) J(nd, 2 p) + J(3 d, 3 p) J(nd, 3 p)) for ' D (6) ( 3 d 8 ( s L ) 3 d 2~ } 3 d 9 ' D ) J 2 ( 2 St + 1) W ( S f 1 f, 4 s') x

\ ^x⁴³(415) R y ( m e 6 M / M 1 M 2 ) ( ~ ( 3 d, 2 p) J(nd, 2 p) + J(3 d, 3 p) J(nd, 3 p)) for ^3~ .

The coefficients which arise from the expansion 3 d 9 4 s + (3 d 8 ) S L (3 d 4 s ) S' L' have been already

given in table 2. The quantities J are defined as calculated using Hartree's functions. They are given

w in table 111. The quantum number n runs from 4 to 6.

J(n1, n' 1 - 1) ⁼a ,

1

^{R(n, 1)}^x Details of the calculation of the excited nd-functions

o are communicated in chapter 5.

1 - 1 The values of table I11 are to be inserted into the

[$

^- ^R(n',^l^-^{l )}^{r 2}^dr^- ⁽⁷⁾ final formulae (8) and (9) compiled from formu- lae (3), (4) and (6). They provide the specific mass 4. Numerical results and comparison with experi- effects of (3 d 9 4 s ) ' D and ( 3 d 9 4 s ) ^3~ as compared ment. - All the quantities in question have been to the ionization limit (3 d 9 ) '0 :

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R0(3 d 4 s, nd 4 s) - - 1 R2(3 d 4 s, 4 snd) 2 J2(2 p, 4 s) + 2 J2(3 p, 4 s) + 2 15

X

Ml M 2 n = 4 &3d - &nd

R0(3 d 4 s, nd 4 s) - 3 1 R2(3 d 4 s, 4 snd)

- -

*

R y [2 ~ ' ( 2 p, 4 s) + 2 J2((3 p, 4 s) + ² ^x

Ml M 2 n = 4 &3d - &nd

Numerical parameters computed with Hartree's functions (3)

R0(3 d 4 s, nd 4 s) R2(3 d 4 s, 4 s nd)

cm-I cm J(nd, 2 P) J(nd, 3 P)

- -

8 912 - 3.80 1.90

(7 950 obs.)

- 6 209 - 0.73 0.15

- 3 611 - 0.44 0.07

- 2 473 - 0.31 0.05

Contributions of the particular terms of formulae (8) and (9)

to the total specific mass effect of the 3 d 9 4 s 'D and 3D terms in Cu 11-spectrum.

Kind of the contribution < 3D 1 J(G) 1 '30 > - < 6(0) >

-

4 s-electron : 16.7 mK

excitation 3 d + 4 d : - 36.1 mK

3 d + 5 d : - 9.1 mK

3 d + 6 d : - 4.0 mK

total : - 32.4 mK

The numerical contributions of the particplar 1. Moreover, the 3 d 9 ID- and 3~-states have different terms of formulae (8) and (9) are listed in table IV. specific mass effect and the calculated value

The results of table IV demonstrate the rise of the < I D 1 1 1 D > - < 3 D 1 1 3 0 > = 3,9 m~

specific mass effect caused by the removal of the

4 s-electron from Cu 11: 3 d 9 4 s-configuartion. Exci- (10)

tation effects of the 3 d-shell overcompensate the direct agrees well with the measured value 4.7 mK. The contribution of the 4 s-electron as expected in chapter agreement is even improved if the higher orbitals

(3) Further computed values to be used in formulae (8) and (9) are J ( 2 p , 4 s) = 0.25 and J(3 p, 4 s) = 0.47.

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EXCHANGE POLARIZATION OF THE 3 d-SHELL A N D ISOTOPE SHIFT C 1 - 4 7

7 d, ..., oo d are taken into account. Then the value of equation (10) increases to an estimated value of 4.2 mK.

5. Numerical calculations of the orbitals 4 s , 4 d, 5 d, 6 d. - Sets of radial eigenfunctions are given for the Cu I I : ls2 2s2 2p6 3s2 3p6 3d1° configura- tion by Hartee [ l o ] and Synek [8]. The functions P(4 s), P(4 d ) , P(5 d ) , P(6 d ) are to be constructed within a potential which roughly agrees with the self-consistent potential of the quoted configuration.

One proceeds as follows. One 3d electron is dropped and from the remaining l s 2 2s2 2P6 3s' 3p6 3d9- electrons the central potential Y ( r ) / r is calculated.

From this potential the function P(n1, r ) was obtained as a solution of the wave equation

where the normalizing condition

was obeyed. Though the exchange interaction was neglected throughout this procedure the agreement of the P(3s, r ) given by Hartree [lo] and that one of our calculation was satisfactory. This demonstrates that P(3s, r ) is very weakly affected by exchange interaction. Using the same potential the function P(4 s, r ) was calculated and normalized.

This procedure failes with respect to the d functions.

The 3d function cannot be reproduced within the potential Y ( r ) / r because the potential is compensated by the centrifugal barrier over a long range. Therefore exchange interaction plays the dominant role. This

difficulty was overcome in the following way [ I l l . The operator

was applied to Hartree's 3d function. The resultant function is divided by the 3d function itself thus providing an effective potential. Within this potential the 3d-function can be reproduced exactly. This potential served for the calculation of the higher nd-functions which thus become orthogonal to each other. These functions served for the calculation of the Slater integrals and the Vinti integrals along with Hartree's 2 p, 3 p and 3 d functions.

Acknowledgements. - Thanks should be given to professor W. Walcher for the sponsorship of this work and to Monsieur J. Bauche, Bellevue, for valuable advices.

References

[I] HUGHES (D.), and ECKART (C.), Phys. Rev., 1930, 36, 694.

[2] BARTLETT (J.) and GIBBONS (J. G.), Phys. Rev., 1933, 44, 538.

[3] VINTI (J.), Phys. Rev., 1939, 56, 1120.

[4] CRAWFORD, GRAY, KELLY and SCHAWLOW, Canad.

J. Phys., 1950, A 28, 138.

[5] ELBEL (M.), FISCHER (W.), und HARTMANN (M.), 2. Physik, 1963, 176, 288.

[6] SCFROEDER (D. J.) und MACK (J. E.), Phys. Rev., 1961, 121, 1726.

[7] BAUCHE ^(J.),C. R. Acad. Sc. Paris, 1966, t. 263 B, 685.

[8] SYNEK (M.), Phys. Rev., 1963, 131, 1572.

[9] ROSENZWEIG (N.), Phys. Rev., 1952, 88, 580.

[lo] HARTKEE (D. R.), und HARTREE (W.), Proc. Roy. Soc., 1936, A 157, 490.

[ l l ] STERNHEIMER (R. M.), Phy.9. Rev., 1954, 96, 951, Phys. Rev., 1957, 105, 158.