Isotope shifts

Chapter 73 Chapter 73

8. Free-ion work since 1965

8.4.3. Isotope shifts

The greater attention being paid to those aspects of hyperfine structure that have to do with atomic physics (rather than nuclear physics) is reflected in studies of isotope shifts. In 1970, Stacey (1971) could speak of using such studies to improve our knowledge of the distribution of charge within the nucleus, a prospect that he cast in terms of the major advances in the theory of nuclear structure that atomic spectroscopists have stimulated in the past. A few years later, Bauche and Champeau (1976), in their classic review of the field, showed how and why interest had moved towards electronic aspects. In the first place, the measurements of Striganov et al.

(1962) on Sm I, which showed that relative isotope shifts are not constant, led to the realization that the specific mass shift (SMS) operator S, given by

S = ~ ~ p,,pj, 1 (111)

i > j

was significant for nuclear masses M as large as those for the lanthanides (King 1963). The calculation of the matrix elements of S is an interesting problem in atomic physics, but there is absolutely no point in using isotope shifts to deduce values of M. In the second place, the distribution of nuclear charge makes itself felt by the electronic field-shift (FS) operator F, which, in the non-relativistic limit, is proportional to Eih(r;). Although the matrix elements of this operator are constant for a configuration - and thus rather uninteresting from an electronic standpoint - their appearance in second-order perturbation theory for a configuration such as 4f%s produces coefficients proportional to those of the Slater integrals

G3(af, 6s)

that occur in the expressions for the term energies. As experimental techniques have improved it has become interesting to examine such extensions of the first-order theory.

The repetition of coefficients just mentioned is not limited to the FS operator. It was Stone (1959) who noticed that the tensorial structure of the term pi. p~ occurring in the SMS operator is identical to that of y}l~. y}l~. The latter appears in a configuration such as 4fN5d in association with the Slater integrals G 1 (4f, 5d). The prefacing coefficients must reappear in a calculation of the matrix elements of S, given in eq. (111). To stress the correspondence, the new SMS parameter is written g~ (4f, 5d). Because of the wide range of operators used in atomic shell theory it is perhaps not very surprising that matched parameters can be found as S and F are taken to second order of perturbation theory. The correspondence can be extended to several contributions to relativistic corrections, thereby introducing spin- dependent operators. The parameters analogous to ~,t are written z,~.

In their review, Bauche and Champeau (1976) gave several examples drawn from

ATOMIC THEORY AND OPTICAL SPECTROSCOPY 165

lanthanide spectra. The usefulness of z4f was illustrated by the isotope shifts for 144-15ZSm 14f66s 2 7F, an example drawn from the 1976 Paris thesis of Sallot. Lines from several transitions of Ce II (for isotope pairs 140-142, 138-140 and 136-140) were used to show how the so-called King plot could be used to separate the effects of FS from those of SMS. Bauche and Champeau (1976) also considered some lines of Sm I and Dy I (where the extent of configuration mixing for the levels involved was known) to show that the experimental SMS was roughly three-fifths of the HF value. Variable agreement with H F calculations was obtained by Wilson (1972) for the experimental FS for SmI,II and EuI,II. The comparison was made via what are known as screening ratios, which measure the ratios of differences of values of

< c l ~ ~(r,)l c )

(112)

for various configurations C. Coulthard (1973) used a relativistic HF program to recalculate (112). He found that substantial enhancements occurred (by factors of 3 or 4), and, although these enhancements largely disappeared when ratios were taken, there was a general improvement in the fit for Sm and Eu previously obtained by Wilson (1972).

Interest in the spectra of S m I and E u I has continued. Bauche et al. (1977) presented new isotope-shift measurements for x44qSZSm and interpreted the J dependence in the levels of 4f66s 2 7Fj in terms of a parameter z4f = 1.68 + 0.30 inK, where l m K = 1 0 - 3 c m -1. This value for z4f corresponds well with the figure of 1.09 _+ 0.22 obtained by Aufmuth (1978)for 162-164DyII 4fX°6s 6I. By taking several pairs of Sm isotopes, New et al. (1981) were able to deduce that the principal contribution to z4f comes from the field-shift operator F taken to second order with the electronic Coulomb interaction. This result is consistent with the H F calculations reported by Aufmuth (1982) for the configurations 4fN6s in 142 144Ndii,

1 5 1 - 1 s 3 Eu II and 162 - ~ 6~ Dy II. The celebrated transitions 4f v 6s 2 ~ 4f 76s6p of Eu I

were re-examined by means of laser absorption by Zaal et al. (1979), who were able to find improved isotope shifts for 151 -153Eu" Griffith et al. (1981) interpreted some anomalies in Sm I as arising from the mutual perturbation of the two close levels at 19 174 and 19 192cm -1 , a phenomenon for which Palmer and Stacey 11982) held F responsible. The parameter zso in E u I 4fV5d6s l°D, 8D was introduced by Pfeufer et al. (1982) and by Kronfeldt et al. (1982); a disconcerting sign difference has now been resolved (Kronfeldt et al. 1984), and there is good correspondence with the Zsd needed for 156-16°Gdii 4fV5d6s SOD (Kropp et al. 1985a). Para- metric analyses of isotope shifts have continued with the work on Eu I 4fV6s6p and E u I 4fV6s7s by K r o p p et al. (1984, 1985b), who have been able to show that the ratio 93(4f, 6s)/G3(4f, 6s) agrees with the H F value. Evidently the second- order corrections to 93 and G 3 are roughly proportional to each other.

8.5. Rydber 9 series

Every configuration involving an electron nl has the potential for generating a

sequence of configurations in which n is replaced by n + 1, n + 2, n + 3, etc. Levels in such a sequence that are matched in some way (usually by selecting a given core state and a given total angular momentum) are said to form a Rydberg series. The rich level structure of many low-lying configurations in the lanthanides almost always leads to a morass of overlapping Rydberg series. Progress has only been possible for cases where the low configurations are particularly simple. G a r t o n and Wilson (1966) identified two Rydberg series of L a I as corresponding to the transitions

6s25d2D3/z.5/2-6s2np2p3/zA/2*

(n = 8, 9 . . . 23) and were able to deduce an improved value for the ionization potential of La I. The relative isolation of 8S in 4f 7 made it possible for Smith and Tomkins (1976) to observe transitions of E u I of the type 4f76s28S-7/2 ~4fV6snp2S+Xpj, where J - 5 ~, 9, and n ~< 62.

Camus and Tomkins (1969)reported the transitions 6s 2 1S 0 --,, 6snpl'3P 1 (n = 6, 7 . . . 48) of YbI, and showed that the smooth trend of the quantum defect #, ( = n - n*, where n* is the effective principal quantum number) was broken by a perturbing level at 4 9 9 2 0 c m - 1 . Laser excitation was used by Worden et al. (1978) to examine the Rydberg spectra of ten lanthanides and to determine ionization limits.

Summaries of the theoretical treatment of Rydberg series have been provided by Fano (1975) and Aymar (1984). Many of the ideas go back to the work of Seaton on multi-channel quantum-defect theory ( M Q D T ) (see Saraph and Seaton 1971). The key element is that each n l electron, for large n, moves over most of its trajectory in a Coulomb potential. Thus the form of the wavefunction at large r is severely limited; in fact, it can be expressed as a linear combination of two basic forms, the relative strength of which fixes the quantum defect. The energies of the levels are given by E = 1 - R / n * 2, where R is the mass-corrected Rydberg constant and I an ionization limit. F o r a Rydberg series, the notion of a channel replaces that of a configuration. The angular momenta of the core and the outer electron are coupled in the usual way, but the radial integration is left in abeyance until channel mixing (the analog of configuration interaction) is considered.

In the late 1970's, the spectroscopy group at the Laboratoire Aimd Cotton (Orsay) returned to the Rydberg series of Yb I (Wyart and Camus 1979, Camus et al. 1980, Aymar et al. 1980, Barbier and Champeau 1980). The bound even-parity spectra (for which J = 0 and 2) and the bound odd-parity spectrum (for which J = 1) were analyzed with M Q D T models involving up to six channels and four limits. The perturbations on the series 4f146snslS 0 (n >/7) and 4f146sndl'3D2 (n >/8) produced by the levels of 4fX46p 2 and 4f135d6s6p could be understood in a detailed way, as could the effect of the levels of 4f135d26s on the series 4 f l % s n p 1.3p1 (n >/12). Sixty levels for which J = 0 could be fit with an RMS deviation of 0.31 c m - 1 ; the corresponding deviations for 125 levels for which J = 2 and 14 levels for which J = 1 were 1.06 and 0.75 cm - 1, respectively. The parameters involved were the eigenquantum defects /a,, the limits Ii for the series i, and the matrix elements Ui~ describing the unitary transformation between the eigenchan- nels ~ and the so-called collision channels i. The details of this analysis are of considerable interest; however, they do not exhibit the spectral features of the lanthanides other than showing that the theory can accommodate the complications

A T O M I C T H E O R Y A N D O P T I C A L S P E C T R O S C O P Y 167

that they give rise to. F o r that reason it does not seem appropriate to take space here to describe the theory at length.

The perturbers of the series 4f~g6sn/of YbI were found (or predicted) by Wyatt and Camus (1979), who used the classic parametric approach to fit the levels of the two mixed configurations

and

4f~4(6s2 + 5 d 2 q- 6p 2 + 5d6s + 6s6d + 6s7d + 6s8d + 6s9d + 6slOd + 6s7s + 6s8s + 6s9s) + 4f~3(6s26p + 5d6s6p + 5d26p),

4f14(6s6p + 6s7p + 5d6p + 6s5f + 6s6f) + 4f13(5d6s2 + 5d26s + 6s6p2).

107 levels of the first configuration and 73 of the second (out of the respective totals 753 and 262) were fitted with deviations varying from a few cm-1 to just over 2 0 0 c m - 1 . Wyart and Camus (1979) were also able to find some missing levels of Y b l I 4f13(5d2 + 5d6p).

The hyperfine structures of xvl'173ybI 4f146snd (24 ~< n ~< 43) were examined by Barbier and Champeau (1980). For these configurations, three terms in the Hamiltonian are of comparable importance: the electrostatic interactions between the 6s and nd electrons, the spin-orbit interaction H~o for the nd electron, and the hyperfine contact term at the nucleus for the 6s electron. The three parameters are GZ(6s, nd), ~,d and as, respectively. As n becomes large the hyperfine parameter a s dominates; for each isotope the configuration 4f146snd possesses a splitting that tends towards that for Y b l I 4 f l % s as the trajectory of the nd electron becomes increasingly remote. Barbier and Champeau (1980) were also able to relate the isotope shifts for the Rydberg series to those of the resonance lines 4f 1%s ~ 4f ~ 46p of Yb II.

Kotochigova et al. (1984) used the H a r t r e e - F o c k - D i r a c method to calculate the energies of the levels of Eu I 4fV(SS)6snp (6 ~< n ~< 17) for which J = 1, and those of YbI 4fn46sn'p (6 ~< n ' ~ < 11) for which J = I. In general, the energies separating pairs of levels with a c o m m o n n or a common n' are underestimated by roughly 20 .... _/o~

though the progression of the configurations to their respective ionization limits is given much more accurately. The underestimates of ff,p and ~,,p indicate too great a radial extension to the p orbitals, a result that is consistent with the overestimate of the matrix elements of r (see section 8,7).

In order to help with the analysis of Rydberg series in EuI, Wyatt (1985)carried out a least-squares fit to the levels of

4f7(8S + 6p)(6s2 + 5d6s + 5 d 2 "q- 6s6d + 6s7d + 6p 2) +4fv(8S)(6s7s + 6s8s + 6s9s + 6sl0s)

by traditional means. 97 levels were fit with a mean error of 58 cm 1. This work also has a relevance for three-step photo-ionization, such as that used to determine the isotope shifts of the short-lived pair 141 - 1 4 3 E u by Fedoseyev et al. 11984). Aymar et

al. (1984) used three-step laser spectroscopy to effect the transitions 4f146s 2 xS o ~ 4f146s6p 3P 1

~4f146s7slSo or 4f146s7s3S 1 or 4f13(2FT/2)6sZ6p312 ( J = 2 ) -~ 4f146snp 1P l, 3p0.1. 2 or 4f146snflF3, 3F2.3,

thereby reaching Rydberg series of odd parity and high n. The effects of perturbers coming from 4f 135d 26s were studied. The revisions made in the earlier analyses for the 6snp series (Aymar et al. 1980) showed how delicate the choice of M Q D T parameters can be. Aymar et al. (1984) concluded that the presence of perturbers near the ionization limit makes it essential to observe many Rydberg series to get a reliable value for the first ionization limit.

Two perturbing levels from 4f135d26s were found by Blondel et al. (1983) to seriously affect the photo-ionization spectra of Yb I from the 6s7s 1S o state in the presence of a static electric field. A more detailed description of this work has been provided by Blondel et al. (1985). Applying static electric fields to free lanthanide atoms is not currently used much. For a recent example of the classic Stark effect, the reader is referred to the measurements of Neureiter et al. (1986) on S m l

(4f66s6p + 4f 55d6sZ)VF1.

8.6. Spectra of highly ionized lanthanides

The importance of astrophysical sources and plasma diagnostics to physicists has meant that there has always been considerable interest in the spectra of lanthanides from which many electrons have been stripped. Atomic species of that kind seldom show features that one associates with the lanthanides; indeed, the spectrum of Eu LXI, should it ever be studied, would bear on the two-electron problem. Such a possibility is not infinitely remote. In table 54 of his review article for the Handbuch der Physik, Edl~n (1964) took the He I isoelectronic sequence only to Ne s +; but, some 17 years later, Martin (1981) was working in a range that extended as far as Ar XVII. The functional dependence of the spectra on Z (the nuclear charge) is of prime interest here. In his extensive review of the spectra of highly ionized atoms reported during the period 1980-1983, Fawcett (1984) cited work on ten lanthanide ions (La XII through Ho XXII) like Pd I, ten ions (La XI through Ho XXI) like Ag I, two ions (Tm XLII and Yb XLIII) like Ni I, eight ions (La XXXI through Yb XLIV) like Co I, eight ions (La XXIX through Yb XLII) like Cu I, and two ions (Tm XL and YbXLI) like Z n I . For the ions like PdI, Sugar and Kaufman (1982) studied the trend of the resonance lines of the type 4d95p3'lp1,3D1---~4dl°lS o and 4d94f 3 D1,1 P1 ~ 4d1° 1 So. They found that there was a sharp increase in both the oscillator strengths and the Slater integrals at the onset of the contraction of the 4f orbit (that is, near Ba XI, where the binding energies of the 4f and 5p orbitals cross).

To fit the energy levels, the H F values for the parameters had to be multiplied by the factors 0.90 for F2(4d,5p), 0.958 for Gk(4d,5p), 0.75 for both Fk(4d,4f) and Gk(4d,4f), 0.85 for

~4f,

1.035 for ~4d, and 1.20 for (Sp. Similar scaling factors were found by Sugar and Kaufman (1980) for lanthanides in the Ag I sequence. As the

ATOMIC THEORY AND OPTICAL SPECTROSCOPY 169 stages of ionization increase, the transitions move to the X-ray region, thus taking them out of the scope of the present review. For example, the 3d-4p, 3d-4f, 3p-4s, and 3p-4d transitions of T m X L I I and YbXLIII are in the region 5-9A (Klapisch et al. 1980). The spin-orbit coupling constants (3d, (4v, (40 and (4f are very large and lead tojj coupling in such configurations as 3d94p, 3d94f, and 3p54d. For a detailed summary of work in the 1970's, the reader is referred to the works of Fawcett (1974, 1981).

8.7. Intensities

The great bulk of the spectra of the free lanthanides consists of lines that correspond to electric-dipole transitions. The contribution to the Hamiltonian is thus proportional to E ' r i for each electron i, and the whole problem turns on evaluating the matrix elements of Ylri. Condon and Shortley (1935) listed the relative strengths of the lines of several transition arrays. The methods of Racah (1942b, 1943) can be readily applied to Ziri. This operator possesses the tensorial characteristics of T I°xn and thus leads to the selection rules AL, AJ = 0, +1;

AS = 0. The details of working out the tensor algebra were described by Levinson and Nikitin (1965), who provided the transitions 4f 7 (ss)6p ~ 4f 7 (sS)6s of Eu II and 4fla6s6p-~ 4fxa(6s z + 6s5d) of YbII as examples.

A more complicated situation was considered by Camus (1970). He compared the data of Komarovskii and Penkin (1969) for 41 lines of T m I with the results of a calculation based on transitions of the type 4fa36s6p + 4fa25d6s 2 --~ 4fla6s 2. The radial integrals were evaluated by the HF method. With one overall scaling parameter, Camus (1970) was able to fit the observed line strengths to within 30~0 on the average. He also checked his working by comparing the lifetimes of twelve excited levels of T m I with the figures obtained by Handrich et al. (1969) on the basis of the Hanle effect. Generally good agreement was found apart from a single level where a marked interference between the components belonging to 4f136s6p and 4f 125d6s 2 occurred.

Sugar (1972) used the calculated intensities for the transitions of the type 4dX°4fs ~ 4d94f N+I (which occur in the soft X-ray region) to confirm that the fami- liar term structure of the 4f-electron configurations appears in the metallic lanthanides. His results were consistent with the trivalent forms R 3 + with the exceptions of Eu 2+ and Yb 2+. The Slater integrals Gl(4f,4d) and F2(4f,4d) had to be reduced from the corresponding H F values by roughly 33'~,, and 25~o, respectively, in order to get a reasonable fit to the experimental absorption curves.

Relative intensities can also be useful in confirming transitions in highly ionized species. Tech et al. (1984) examined the 4dg-4d85p transition array in L a X I I I (which occurs in the region 90-110A) and they were able to correlate 31 missing lines with low predicted intensities. The 25 known levels of L a X I I I 4d85p were fit with nine adjustable parameters to an R MS error of 154 c m - a . This is only 0.1'}o of the energy spread of the configuration.

Absolute line strengths are of considerable interest because a direct comparison can be made with the results of H F calculations. Dohnalik et al. (1979) measured the

oscillator strengthsf of three lines of Er I 4 f 12 3H6 (6S 2 __~ 6s6p) with quoted errors of 25~, and found them roughly half as large as the corresponding HF values.

Szynarowska and Papaj (1982) measured f f o r a single line of Gd I, thus scaling the 138 relative f values found by Penkin and Komarowsky (1973)for that spectrum.

The latter authors were able to find the absolute values of f for the two lines 4f14(6s 2 -* 6s6p 3P 1, 1P1) of YbI as 0.014 and 1.12, respectively (Komarovskii and Penkin 1969), which they compared to the figures 0.0167 + 0.008 and 1.30 + 0.06 obtained by Baumann and Wandel (1966) from the lifetimes derived by means of the zero-field crossing technique. A later measurement by Gustavsson et al. (1979) using the pulse modulations of a CW dye-laser beam yielded f = 0.0159 + 0.005 for the first of the two transitions, thus indicating that the figure of Komarovskii and Penkin (1969) may be somewhat too low. Similar discrepancies occur for E u I (Gustavsson et al. 1979, Komarovskii et al. 1969). However, the differences between the experimental results are small compared to their deviations from the ab initio calculations. Loginov (1984) evaluated the radial transition integrals for EuI 4fT(aS)6s(5d ~ 6p) using the Hartree-Fock-Dirac method and found them to be two to three times larger than those deduced directly from experiment. It is somewhat disconcerting, to say the least, that better results are obtained by the Coulomb approximation (Loginov 1984). That method is empirical by nature, being based on Heisenberg's form of the correspondence principle for non-relativistic matrix elements (Naccache 1972) and developed with considerable ingenuity by Picart et al. (1978) and Edmonds et al. (1979).

Dalam dokumen Handbook on the Physics and Chemistry of Rare Earths (Halaman 170-176)