Dipole polarizabilities from the nf term values

NUCLEAR

INSTRUMENTS

AND

METHODS

IIO

(I973)

241-244;

©

NORTH-HOLLAND

PUBLISHING

CO.

D I P O L E P O L A R I Z A B I L I T I E S F R O M T H E nf T E R M V A L U E S * P. V O G E L

California Institute of Technology, Pasadena, California 91109, U.S.A. Experimental term values o f the nf series have been used for an empirical determination o f the dipole polarizabilities ~a for a n u m b e r o f ions in the Ne 1, N a I, a n d A 1 isoelectronic series. The core charge densities obtained by the relativistic H a r t r e e Fock-Slater m e t h o d were used for the calculation o f the pene-

tration correction. Other corrections related to the incomplete separation o f the electron from the core are also discussed. ~d, unlike the coefficient o f ( r 6) (usually interpreted as q u a d r u pole polarizability c%) is relatively insensitive to the uncertainties o f the procedure.

1. Introduction

leading to eq. (1) very well. We shall show, however, that the parameter So can be determined from the nf series with reasonable accuracy using the relation (1). The first obvious correction to the formula (1) is a penetration correction, i.e., the effect of the finite distribution of the core charge. We have used

A number of hydrogen-like transitions were recently identified in the beam-foil spectra of several elements~'2). For their proper interpretation and for correct understanding of the processes leading to their excitation, it is vital to know the dipole polarizabilities sd of the corresponding ion cores. The deviations of the observed transition energies from the purely hydrogenic values are small and determined with relatively large error bars. Thus even approximate values of sa would be helpful. The difficulties and complexities of the theoretical determination of the atomic dipole polarizabilities were described by Dalgarno3). Among the empirical possibilities of the s d determination, the most adequate for the present purpose seems to be the method using the term values of non-penetrating electronic orbits [as described in detail, e.g., by Edl6n4)]. This procedure is based on the expression

AT

=

T e x p - - T n = Sd R y ( r - 4 )

+ Sq R y ( r - 6 ) ,

(1)

where Texp and T n are the experimental and theoretical (relativistic) hydrogen-type term values. The parameters So and Sq are interpreted as dipole and quadrupole polarizabilities. If several term values are known, the cq and Sq parameters are determined from the linear relation (AT/Ry) ( r - 4 ) -1 = s c i + S q ( r - o ) / ( r - 4 ) .

(la)

2. Dipole polarizability and the nf series

For multiply-charged ions with 10 ~ Z ~ 32, the states with I > 4 are seldom known. On the other hand, the electron nf states do not fulfill the conditions * This work was p e r f o r m e d u n d e r the auspices o f the U. S. A t o m i c Energy C o m m i s s i o n . Prepared u n d e r Contract AT(04-3)-63 for the San Francisco Operations Office, U.S. Energy C o m m i s s i o n .

Ape, = T , t - T . ,

(2)

where T,t is the eigenvalue of the Dirac equation with the potential of the extended core. The similar quantity calculated using the first-order perturbation method is somewhat (not more than 15%) smaller. The core potential Vc(r) and the core charge density p(r) were obtained by the relativistic Hartree-Fock-Slater method. All ions were assumed to be in their ground states. Several Ap~. values are shown in column 5 of table 1. Note the fast increase of Ape n with the ion charge; nevertheless, even in Co IX, the penetration is only about 1% of the binding energy and exchange interaction can still be neglected. The similarity of the Ape n and ( r - 6 ) dependences on the quantum number n (column 6, table 1) implies that penetration strongly affects the quadrupole parameter Sq, but has a much smaller effect on the parameter s d and on the linearity of the dependence (la). The incomplete separation of the nf electron from the core leads to a modification of the ( r -4 ) and ( r - 6 ) factors in eq. (1) as well. Let us consider the dipole ( r - 4 ) part. The electric dipole interaction between the electron [wave functions ~po(r), ~0k(r)] and the core [wave functions ~90(R~), ~m(R¢)] is VE 1 =

--e 2 Re' r r3 ,

= _e2 Rc.r

g 3'

r >

Re,

r < Re.

(13)

Using the second-order perturbation theory, neglecting

241 III. T H E O R Y

242

P. VOGEL TABLE 1

Term values (Texp), polarization (dpoD and penetration shifts (ripen), and dvalues. ~ is the ion charge. The quantitiesdpoh ripen, and d () are defined in eqs. (7), (2) and (6), respectively.

Ion

State

Texp (cm -1)

Apol (cm t)

,dpen (cm-1)

~2 Ape. Ry

Ryd () ~4

~2A ()



Ca 1I

4f 5f 6f 7f 8f

27 695.0 17 717.5 12 293.8 9 024.8 6 904.6

250.2 150.9 94.6 62.3 42.9

10.5 8.7 6.2 4.4 3.2

7.75 8.74 9.19 9.42 9.59

0.0351 0.0298 0.0217 0.0156 0.0112

0.412 0.476 0.512 0.536 0.548

Ge II

4f 5f 6f 7f 8f

28 202.8 18 015.6 12 478.8 9 146.1 6 988.0

737.9 433.1 268.6 176.0 121.0

30.7 24.6 17.2 12.0 8.3

22.56 24.63 25.40 25.82 25.27

0.232 0.192 0.137 0.097 0.069

2.72 3.08 3.24 3.33 3.36

Co IX

4f 5f 6f 7f 8f

578 710.0 370 670.0 257 330.0 188 280.0 143 800.0

14 585.0 8 848.0 6 242.0 4 048.0 2 936.0

8 498.0 6 220.2 4 144.0 2 806.0 1 962.0

15.23 15.18 14.93 14.73 14.58

0.599 0.524 0.381 0.273 0.197

7.05 8.34 8.99 9.32 9.56

Cu XIX

4f 5f 6f 7f

4 169.0 2 771.0 1 299.0 1 169.0

2 310.0 1 767.0 1 208.0 832.0

0.984 0.752 0.514 0.354

11.55 12.06 12.13 12.17

2 476 1 589 1 103 810

671.0 740.0 360.0 784.0

t h e e l e c t r o n c o n t r i b u t i o n to t h e e n e r g y d e n o m i n a t o r s a n d n e g l e c t i n g the r
<+o.+>. i<

/=

= - ½ e2 ~d ( r - 4 > •

(4)

I f t h e r i n s t e a d o f ( r - 4 > in eq. (1). T h e f o r m f a c t o r f(r) was c a l c u l a t e d a p p r o x i m a t e l y here, u s i n g the l o w e s t d i p o l e excited state f(r)=

If 1-

~o(R~-ra)¢.dRo

0.208 0.217 0.219 0.220

0~q. T h e r e l a t i v e insensitivity o f the t~d v a l u e to t h e inclusion of the penetration a p p r o x i m a t i o n u s e d f o r
L5Ir

1~

i\

li!

i

,o?,! i \

/;o

~oR~.dR¢ (5)

c o r r e c t i o n a n d to the > a n d is illusf r o m the s t r a i g h t line axis (i.e., the ~d value)

.5-

,-

\~,. \v"

~.~

¼,

/I/ ~.,

[an e x a m p l e o f t h e f u n c t i o n f(r) is s h o w n in fig. 1]. C o l u m n 7 o f t a b l e 1 gives the q u a n t i t y p r o p o r t i o n a l to A((r-4>)

~ < r - 4 > --

(r-4f(r)>.

(6)

As is apparent from the last column of table 1 that A ( < r - 4 > ) varies with n similarly to and therefore primarily affects the quadrupole parameter

2

4

6

8

10

r Fig. 1. Core charge density p(r), form factor f(r), and square of the 4f and 5f wave functions for Co IX (18e core). Radius r is in 10-9 cm, ordinate in arbitrary units.

243

DIPOLE POLARIZABILITIES

are almost the same for the four lines shown. A similar conclusion is valid in all considered cases. The similarity of the Ape,, A((r-4)) and ( r - 6 ) dependence on n is not an accident. The corresponding integrals get in all three cases largest contributions from the same region of r. 3. Results and discussion

The list of ions considered is given in table 2. A modified eq. (1) Apol -- T e x p - T H - A p e n

=

~d

RY(r-4) +

(~q R Y ( r - 6 )

(7) was used for the c% determination. The resulting dipole polarizabilities are given in column 4. The parameters ~q obtained from the fit of eq. (7) cannot really be interpreted as the quadrupole polarizability. As is obvious from the preceding considerations a number of effects contribute to the coefficient of < r - 6 > . According to OpikS), nonadiabatic effects contribute basically to the coefficient of ( r - 6 ) as well. The hydrogenic values [given, e.g., by Edl6n4)] were used for the ( r - 4 ) and ( r - 6 ) in eq. (6). We felt that using the form factor f(r) (or a cut-off radius) would not make the results more reliable. Nevertheless, the differences between the (~d values derived from eqs. (7) and (1) or from eq. (7) with (r-4f(r)) instead

O

Fe XVI Co X V I I Ni X V I I I Cu X I X Mg I Si I I I K VIII Ca IX Sc X TiXI AII Si II K I Ca II Mn VII Fe V I I I Co IX Fe Ili Ga I Ge II a b e d e f

.5

I I

Core States included

10e 10e 10e 10e lie 1 le 1 le 1 le lie lie 12e 12e 18e 18e 18e 18e 18e 23e 30e 30e

4 f - 8f 4 f - 7f 4 f - 7f 4 f - 7f 4 f - 12f 6h-9h, 7i-9i 4 f - 7f 4 f - 6f 4f-6f 4f-6f 4 f - 8f 5g- llg 4f - 9f 4f-10f, 5g-9g 4f-9f 4 f - 7f 4 f - 9f 4f,5f,5g,6g,6h 4 f - 8f 4f-8f, 6g, 7g

Ctd(a0a)

Reference

0.0038+0.0010 0.00244-0.0010 0.0040-4-0.0015 0.002 4-0.001 33.0 4-0.5 6.86 0.55 4- 0.07 0.47 4-0.06 0.40 4-0.10 0.31 4-0.07 24.3 11.2 5.47 3.12 0.29 4-0.15 0.37 4-0.07 0.26 4-0.04 1.67 4-0.08 18.14 10.0 4-0.2

a a a a b e d d a d e f g h f f i J k

U. Feldman et al., J. Opt. Soc. Am. 61 (1971) 91. G. Risberg, Arkiv Fysik 28 (1965) 381. y . G. Toresson, Arkiv Fysik 18 (1960) 389. j. O. Ekberg, Phys. Scripta 4 (1971) 101. K. B. S. Eriksson and H'. B. S. Isberg, Arkiv Fysik 13 (1963) 527. Ch. E. Moore, Atomic energy levels, Nat. Bur. Std. (U.S. G o v e r n m e n t Printing Office, Washington, D. C., 1949). P. Risberg, Arkiv Fysik l0 0956) 583. B. Edl6n and P. Risberg, Arkiv Fysik 10 0956) 553. E. Alexander et al., J. Opt. Soc. Am. 55 (1965) 651. S. Glad, Arkiv Fysik 10 (1956) 291. I. Johansson and U. Litzen, Arkiv Fysik 34 (1967) 573.

A

y

0

Ion

g h i J k

1.0

.2

TABLE 2 List of ions included in the calculation.

] 2

I 3

I 4

[ 5

of ( r - 4 ) were used for a crude estimate of the uncertainty in ~d. The experimental quantity Texp depends not only on the accurately measured wavelengths but also on the value of the series limit. For the multiply charged ions only a few members of a series are usually known and the series limit determination is often difficult. To exclude contradictory experimental data (or disturbed series), the consistency of published term values was tested. The series limit and the coefficients of the linear Ritz formula

×

Fig. 2. Dependence o f y = d T/Ry (r -4) on x = (r-6)/(r -4) for the nf states in Co IX. Points ( ~ were obtained using eq. (1) with hydrogenic ( r - 4 ) and ( r - 6 ) values. Points x are from eq. (7). The (r-4f(r)) were used instead o f ( r - a ) to obtain the points A. Finally, [] points were obtained by using eq. (7) and penetrating eigenfunctions for ( r - a ) and ( r - 6 ) .

6 = n - x / ( R y ~2/T) =

a+bT

(8)

were determined by the least-square method from the term values of the nf series. The term values which flagrantly violate eq. (7) were excluded. Besides, we expect the quantum defect ~ to vary smoothly with III. THEORY

244

P.

102

~×~'i2e

F e V - F e V I I I (ref. 7) a r e in t h e i n t e r v a l 0.5 _< ~a < 1.0, in g o o d a g r e e m e n t w i t h o u r e x p e c t a t i o n . H o w e v e r , t h e p o l a r i z a b i l i t y o f F e V I I I c o r e is s o m e w h a t l a r g e r than our prediction-suggesting that the 18-electron c o r e is n o t in its g r o u n d state. The results have shown that the nf term values can be u s e d f o r ~d d e t e r m i n a t i o n in t h e c o n s i d e r e d r e g i o n o f Z . T h e % d e r i v e d f o r m u l t i p l y - c h a r g e d i o n s a g r e e reas o n a b l y well w i t h e x t r a p o l a t i o n b a s e d o n eq. (9). T h e s c r e e n i n g c o n s t a n t s ( Z ) is a s l o w l y d e c r e a s i n g f u n c t i o n o f Z ; it is p r a c t i c a l l y c o n s t a n t f o r i o n c h a r g e > 10. T h e a s y m p t o t i c v a l u e s o f t h e s c r e e n i n g c o n s t a n t s ( Z ) o b t a i n e d h e r e a r e s ---- 11.4 f o r t h e 1 8 - e l e c t r o n c o r e , s = 5.3 f o r t h e 1 0 - e l e c t r o n c o r e , a n d s = 7.55 f o r t h e 1 l - e l e c t r o n c o r e , w i t h e s t i m a t e d e r r o r -t-0.10

f×

lOr

C~d

VOGEL

,~

~

30e

I%

~×

T

References

lOe

I(5 2

1(55

1) s. Bashkin and 1. Martinson, J. Opt. Soc. Am. 61 (1971) 1686. 2) W. N. Lennard, R. M. Sills and W. Whaling, Phys. Rev. A6 (1972) 884. a) A. Dalgarno, Advan. Phys. 11 (1962) 281. 4) B. Edl6n, Handbuch der Physik, vol. 26 (ed. S. FliJgge; Springer Verlag, Berlin, 1964) p. 80. 5) U. 6pik, Proc. Phys. Soc. 92 (1967) 566. 6) M. E. Zohar and B. S. Fraenkel, J. Opt. Soc. Am. 58 (1958) 1420. 7) W. N. Lennard and C. L. Cocke, this conference.

l

I0

II

12 ; 14 16 18 20 22 24 26 28 30 32 13 15 17 19 21 23 25 27 29 31 z

Discussion

Fig. 3. Dipole polarizabilities ~d in units of a03. The ~fi calculated in this work are denoted by x ; those from ref. 3 by A, and the values from ref. 4 are denoted by O . The error bars reflect both the uncertainty in the series limit and the range of the cql values obtained using different approximations described in the text. t h e i o n c h a r g e 3. U s i n g t h e s e c r i t e r i a , t h e d a t a o f N i X , C u X I (ref. 6) a n d C r V I (ref. i in t a b l e 2) w e r e excluded. All dipole polarizabilities determined here, together w i t h a few m o r e f r o m refs. 3 a n d 4, a r e s h o w n in fig. 3. The curves were obtained by a smooth extrapolation of more reliable low-charge values of the screening c o n s t a n t s(Z) in t h e f o r m u l a

~d = C / [ Z - s(Z)-] 4 .

(9)

T h e ~d v a l u e s d e r i v e d f r o m t h e b e a m - f o i l s p e c t r a o f

GARCIA: Did your last comment imply that you had a 2-electron excitation in that particular situation ? VOGEL: I do not know. l calculated the dipole polarizability of Fe IX and l got a value of 0.35 or something like that. Then Lennard measured the hydrogenic states in the same core and applied eq. (7), without the quadrupole term, which is unimportant. From that one can calculate the ~d, and he got something like 0.5, which is somewhat bigger. So what does it mean? 1 presume it means that the core is excited, or at least most of the time it is excited. GAkCtA: SO that does mean a 2-electron excitation, so it could Auger then and most likely will instead of radiating. VOGrL: | do not know. COCKE: You will see tomorrow some of the spectra from which these polarizabilities come but the shifts in energy from which they are taken are mean shifts. There is structure in the spectral lines and it is not at all clear that the polarizability description just by itself can account for the entire spectral shape of the lines, so there is a lot more going on than meets the eye. VOGEL: I certainly agree with that.