Theoretical treatment of f-state spectra

0 (Degree)

J. V. BEITZ

3. Theoretical treatment of f-state spectra

actinide series to Am 3+. For Cm 3+ and heavier actinides, molar absorptivities become more or less constant but continue to remain larger, in general, than for heavy lanthanides.

174 J.V. BEITZ

Trees parameters) with L being the total orbital angular momentum. G(G2) and G(Rv) are Casimir's operators for groups G 2 and

R7,

respectively. Three-body operators are denoted by t i with corresponding parameters, T i. Two-body pseudo-magnetic operators are denoted as p: with corresponding parameters, P:. The Marvin integrals, M h, represent spin-spin and spin-other-orbit relativistic corrections. The last summation in the above equation represents the crystal-field interaction in which Bkq are crystal- field parameters and ckq are spherical tensors of rank k that depend on the coordinates of the ith electron. The summation involving i is over all f electrons of the ion of interest. Equation (1) is used to calculate the expected energies of crystal-field split states. The values of parameters in eq. (1) are adjusted using a nonlinear least-squares method to obtain the best agreement with experimentally determined and assigned crystal-field levels.

The number of experimentally determined and assigned levels often exceeds the number of free-ion and crystal-field parameters by several fold. Even so, local minima are a significant problem in the least-squares adjustment procedure. The practice is to allow only a fraction of the parameters given in eq. (1) to vary freely. Note that fitting of model free-ion parameters to experimental data recorded in condensed phases results in parameter values that differ from the free-ion values found in atomic spectroscopy studies of lanthanide ions (Goldschmidt 1978) and actinide ions (Freed and Blasse 1986) in the vapor phase. For this reason, the term "free ion" is used to refer to parameters or energy levels determined from studies on ions in condensed phases.

3.2. Comparison of "free-ion" states and observed spectra

The energy level differences between 4f states of aquated trivalent lanthanide ions (i.e., the experimentally determined centers of gravity of observed 4f-4f bands), are quite similar to those deduced from studies of the same ions doped into LaC13 (Dieke 1968, Morrison and Leavitt 1982) or LaF 3 (Carnall et al. 1989) as Carnall (1979) stressed in his review of lanthanide-ion spectroscopy. Expressed in another way, there is little difference between the energy-level structure of "free-ion" 4f states of trivalent lanthanides doped into LaC13 or LaF 3 and the 4f-state energy-level structure deduced from spectral studies of aquated trivalent lanthanide ions (e.g., see Sinha 1983).

A similar situation occurs for 5f bands of aquated trivalent actinide ions in that their centers of gravity occur at energies quite similar to the energy differences between the corresponding 5f-state multiplets of the same actinide ion doped into LaC13.

A graphic comparison of available aquated trivalent lanthanide- and actinide-ion spectra and "free-ion" f-state energies is made in figs. 1-21. These figures show the calculated f-state energies as short, vertical lines above the absorption bands. Where a significant energy gap exists between two f-states, the J value of the higher-lying of the two states is indicated (J is the quantum number associated with the total angular momentum of all the electrons in the ion). For ions having an odd number of f electrons, the value shown is J + ½. The "free-ion" energies displayed in figs. 1-21 are those determined in systematic interpretations of the spectra of trivalent lanthanide ions doped into LaF 3 (Carnall et al. 1989) and actinide ions doped into LaC13 (Carnall 1992). Tables 1 and 2 list the values of the "free-ion" parameters used to calculate the f-state energies shown in figs. 1-23. Good agreement between the calculated "free-ion"

I "

..-,

¢-q

e q ] I

+ ~

~ ~ I ~ I

1 7 6 J . V . B E I T Z

"=!

- , ~ 2

; ~ eq

+ ~ .

I ~ I

I I

I ~ I

P I

f-states of trivalent lanthanide ions and their observed aquated ion f - f absorption bands is visually evident when considering isolated "free-ion" states and their associated absorption bands. Strong absorption at ultraviolet wavelengths is often attributed to f ~ d transitions although in the case of easily oxidized ions, such as U 3 +, metal-to-ligand charge-transfer states likely contribute as well. In many cases, particu- larly for the light actinides, it is clear that the f states occur in such close proximity that the observed absorption bands consist of contributions from several f states. In a few of the near-infrared absorbing bands of aquated U 3 + (fig. 3), some systematic devi- ation evidently occurs between the calculated "free-ion" states and the centers of gravity of absorbing bands.

It may happen that a predicted band is too weak to be evident in recorded absorption spectra. This is the case in fig. 9 for the Am 3+ 7 F o ~ S D 1 band whose center of gravity is calculated to occur at 17271cm -1. This absorption band is expected to be very weak on theoretical grounds (Carnall and Wybourne 1964).

However, confirmation of the energy of the 5D 1 state is available from fluorescence studies on aquated Am3+(Beitz et al. 1986, Yusov 1990) and on Am 3+ in a heavy- metal glass at 10 K (Valenzuela and Brundage 1990) that position the 5D1 state within 100 c m - 1 of its calculated energy. The absorption bands of Es 3 + that are expected to be centered at 9701 and 11 963cm- 1 (see fig. 17) are additional instances in which no absorption band is evident at the calculated "free-ion" state energy. Carnall et al.

(1973) presented intensity calculations indicating that these bands should be very weak in absorption; experimental data on Es 3 + doped into LaC13 at low temperature have identified all of the expected crystal-field components of both of these states (Carnall 1992). These Am a+ and Es 3+ examples serve as reminders of the synergism that has occurred between energy-level structure and transition-intensity calculations for trivalent lanthanide and actinide ions. In addition, analysis of transition intensities can provide a basis for quantifying the relative contributions of individual "free-ion"

states to an observed absorption band and so further aid in assigning aquated ion spectra.

The entire energy range of calculated "free-ion" f states is shown in fig. 22 for trivalent lanthanide ions and in fig. 23 for trivalent actinide ions. Several features are evident in comparing figs. 22 and 23. Actinide ions tend to have a somewhat more open energy-level structure near their ground states but a less open energy-level structure for higher-lying states. This arises from larger spin orbit coupling parameters, but reduced Slater parameter values in the actinide series when parameter values for actinide and lanthanide ions having the same number of f electrons are compared (see tables 1 and 2). For mid-series lanthanide and actinide ions, it is evident that many or even most f states occur at energies corresponding to vacuum-ultraviolet or shorter wavelength light (i.e., at energies where little observed data on f-state energies exist). The present lack of relevant experimental data prevents assessment of the accuracy of the calculated energies for such very high lying f states.

3.3, Transition intensities

Using a theory independently developed by Judd (1962) and Ofelt (1962), observed strengths of absorption bands can be understood systematically in terms ot ~ a param-

178

200000

J.V. BEITZ

. 2 . °

150000

~ m

= i-

. ~ 1 0 0 0 0 0

E = _ i ~ -

t -

m m ~ - =

50000 i

- I !

0 -- ~ "r---~

Ce 3+ pr3+ Nd3+ pro3+ Sin3+ Eu3+ Gd3+ Tb3+ Dy3+ Ho 3+ Er3+ Tin3+ yb3+

4f 1 4f2 4f 3 4f 4 4f5 4f6 4f7 4f8 4f9 4f10 4fll 4fl2 4f13 Fig. 22. Calculated "free-ion" energy-level structure of 4f states of trivalent lanthanide ions [energy-level data from Carnall et al. (1989)].

200000

.O E

(-.

q.i 0

150000

100000

50000

^~ ^m _{- i i} | _i ^{m !} =ii!"

^{- =}

_{n "} ^__~

_-

pa3+ U3+ Np3+ pu 3+ Am3+Cm3+ Bk~3+ Cf3+ Es3+ Fro3+ Md3+ No3+

5f 2 5f 3 5f4 5f5 5f6 5f 7 5f8 5f9 5fi0 5f1! 5f12 5f15 Fig. 23. Calculated "free-ion" energy-level structure of 5f states of trivalent actinide ions Fenergy-level data from Carnall (1989b)].

eterized model. This model has come to be termed the Judd-Ofelt model. The oscillator strength, P, of an absorption band is defined as the area under the band envelope as normalized to the concentration of the absorbing ion and the path length of the light through the absorbing medium. The experimentally determined oscillator strength can be related to electric- and magnetic-dipole contributions by the relationship (Condon and Shortley 1963)

81r2mca

[(n2 + 2)2 F2 + n]~21 (2)

P - 3heZ(2J +

1) 9n

where m is the mass of the electron, c is the speed of light, a is the energy of the transition, h is the Planck constant, e is the charge of the electron, _P and 3/3" are the electric-dipole and magnetic-dipole matrix elements, respectively, that connect the initial state, J, to the final state, J; and n is the index of refraction of the medium.

Only a few observed transitions of 3 + lanthanides and actinides have any significant magnetic-dipole character. For such transitions, the /~2 values can be obtained from knowledge of the eigenvectors of the initial and final states. The magnetic-dipole contribution to the oscillator strength of f - f absorbing transitions has been calculated for 3 + lanthanide ions (Carnall et al. 1968a) and 3 + actinide ions (Carnall 1989b). Judd-Ofelt theory has been used to compute ff 2 via the relationship

Fz = e2 ~ -Ok(~J[[ U(k)[t

~,j,)2,

(3)

k = 2 , 4 , 6

where

U (k)

is a unit tensor operator of rank k and -Ok are three parameters that, in practice, are evaluated from measured band intensities. On theoretical grounds, these parameters involve the radial parts of the f-shell wave functions, the wave functions of interacting configurations such as those of higher-lying d and g states, and the interaction of the metal ion and the ligands surrounding it.

Interest in obtaining Judd-Ofelt model parameters often arises when intensities are required for transitions not ordinarily observable in absorption, such as from a fluor- escing state to a lower lying state that is not the ground state. In fluorescence, the purely radiative rate of relaxation of an excited state, (qJJ), to a particular lower state, (tp, j,), is the Einstein A coefficient, which, following Axe (1963), can be expressed as

64n4cr3

[n(n2+2)zff2+n3f/121

⁽⁴⁾

A ( ~ J , ~ ' J ' ) -

3h(2J + 1) 9

where the terms are as defined in eq. (2) above.

Transition-intensity analysis has proven to be a useful adjunct in assigning the states giving rise to spectrally overlapped absorption bands oflanthanide and actinide ions (see e.g., Carnall et al. 1984). Such bands are usually resolved into their constitu- ent absorbing transitions by assuming that the underlying lines follow a modified Gaussian Lorentzian form described by Carnall et al. (1968a). Analyses of this type have resulted in values of I2 k for all aquated trivalent lanthanides (Carnall 1979) and many aquated trivalent actinides (Carnall and Crosswhite 1985a, b). These values are graphically compared in fig. 24. As Carnall (1979) observed, the ,O h values for trivalent

180 J.V. BEITZ 250

200

~-~ 150

0 c)

e~ 100 0 I

~ 50

I I

U Np

',Pu

,' ,, = . Bk Cf

- - Q - • •

Er A = Q 2 0 = ~ 4 : ' [ 3 = ~6

t~ Q

Tm Yb

P J

12 14

Pr Nd Pm Sm Eu Gd Tb Dy Ho

--50 I ~ I ~ I I I

2 4 6 8 10

n u m b e r of f elecfrons

Fig. 24. Comparison of reported Judd-Ofelt theory parameters for aquated trivalent lanthanide ions (open symbols) and actinide ions (solid symbols, error limits shown, data points connected by broken line).

Lanthanide data are taken from Carnall (1979), actinide data from Carnall and Crosswhite (1985b), based on fixing the value of g22 at 1 x 10-2°cm 2.

lanthanides become essentially constant for elements heavier than Nd. A similar effect may be discerned in the actinide values beginning at Cf. Based on trends in O k values, Carnall (1986) has argued that Am 3 + and Cm 3 + probably play roles similar to Pr a ÷ and Nd 3 + among lanthanide ions. The comparatively large estimated errors for light actinide O k values result from fitting to a small database (i.e., from the difficulty in determining oscillator strengths for more than a handful of 5f states for light actinides). In a qualitative sense, the large O k values for light actinides can be attributed to the interaction of 5f electrons of these ions with d-electron states (CarnaU 1986).

3.4. Hypersensitive bands

Although trivalent lanthanide- and actinide-ion f - f transitions usually are regarded as being insensitive to the nature of coordinated ligands, a few absorption bands of such ions do exhibit distinct sensitivity. These transitions satisfy the same selection rules as electric-quadruple radiation and have come to be termed hypersensitive transitions. Judd (1962) noted that large values of (~eJll

u<2)ll ~,j,)2,

^generally

symbolized as U (2t, correlated with lanthanide-ion bands whose intensity differed significantly between chloride and nitrate solutions. Other early theories concerning this effect are presented by Peacock (1975). Henrie et al (1976) found a linear correlation of ligand basicity with oscillator strength for hypersensitive transitions of lanthanide ions and suggested a mechanism for hypersensitivity involving metal-ligand covalency via charge-transfer levels. Misra and Sommerer (1991) have reviewed

absorption spectra of lanthanide ions with an emphasis on hypersensitive transitions as well as transitions of P r 3 + and Nd 3 + that arise from what these authors term

"ligand-mediated pseudo-hypersensitivity". The literature is not in agreement as to which ~ f bands should be considered hypersensitive [compare the entries in table 2 of Peacock (1975) to those in table 1 of Henrie et al. (1976)]. Carnall et al. (1968b-e) list U (2) values for Pr 3+ through Yb 3+. Values of U (2) for 5f-5f transitions of the actinide ions from U 3+ through Md 3+ have been reported by Carnall (1992) in his analysis of the spectra of trivalent actinides in LaC13. Table 3 lists the term symbol, transition energy (based on "free-ion" energies), and U ~2) values (if larger than 0.01 or hypersensitive) for trivalent lanthanide and actinide ions.

If one excludes states that give rise to absorption bands at 1.4~tm or longer wavelength (i.e., states obscured by solvent absorption in H20 solutions) and states of Pm 3+ (rarely studied because of the radioactivity of Pm isotopes), it is evident from table 3 that a U ~2) value of 0.2 or larger for a lanthanide-ion absorption band correlates with that band being hypersensitive. No experimental absorption spectra are available for Fm 3+ or Md3+; U 3+ is so easily reduced that studies on it in aqueous solution are very difficult and few have been reported. Assuming that one also can exclude the 562 state of Np 3 + and the 6H13/2 state of Cf 3 + (on the grounds of scarcity of reported solution absorption studies on these ions), then the same relationship holds for trivalent actinide ions: a U ~2) value of 0.2 or larger correlates with that actinide-ion band being hypersensitive. As is evident from the data shown in table 3 for the bands of Pr, Nd, Sin, Eu, Dy, and Am that are considered to be hypersensitive by Henrie et al. (1976), a value of U ~2) < 0.2 does not preclude a band from being hypersensitive. For ions having the same number of f electrons, the data in table 3 show considerable correlation between a lanthanide S L J state that has a larger U ~2) value and the magnitude of the U ~2) value of the corresponding actinide S L J state. For example, the state of Dy 3 + (a 4f 9 ion) that has largest U ~2) value is 6F11/2; the state of the corresponding 5f 9 ion, Cf 3 +, that has the largest U (2) value is also 6Fll/2. Judd (1988) commented on the status of theoretical efforts to explain hypersensitivity. He suggests that the conclusion of Peacock (1975) that various mechanisms combine or interfere to different degrees under varying experimental conditions may still be valid.

3.5. Vibronic bands

Spectral bands of an aquated lanthanide ion arising from vibronie contributions were reported first by Haas and Stein (1971) in their study of the emission spectrum of aquated Gd 3 +. These bands are termed vibronic because they arise from a simulta- neous change in the electronic state of the metal ion and the vibrational state of a coordinated ligand. Stavola et al. (1981) noted additional examples of such bands and presented a theoretical model based on the importance of electronic factors for calculating the intensities of lanthanide-ion vibronic transitions. Their theoretical model also predicts selection rules for such transitions. The intensities of observed bands assigned by these workers as being vibronie typically were at least 50times weaker than the parent purely electronic band. Faulkner and Richardson (1979) have

182 J.V. BEITZ TABLE 3

Listing of absorption bands with calculated U (2) > 0.01 for aquated trivalent lanthanide ions (Carnall et al.

1968a-e) to 50000cm -1 and trivalent actinides in LaC1 a (Carnall 1989b) a to 31 000cm 1 with data for experimentally ohser'¢ed hypersensitive bands (Henrie et al. 1976) b shown in bold type.

Transition Transition

Lanthanide Excited energy Calculated Actinide Excited energy Calculated

ion c state (cm- 1) U~2) ion ~ state (cm- 1) U~2)

p r 3+ p a 3+d

(3H4) 3H s 2322 0.1095

3F 2 5149 0.5089

3F 3 6540 0.0654

3F 4 6973 0.0187

1D2 16 840 0.0026

3P 2 22 535 ~ 0

Nd 3 + U 3 +

(419/2) 4111/2 2007 0.0194 ('I9/2)

4G5/2 17 167 0.8979

2G7/2 17 333 0.0757

4G7/2 19 103 0.0550

p m 3 + Np3 +

(514) 515 1577 0.0246 (514)

5G z 17857 0.7215

5G 3 18 256 0.1444

3G 3 21 102 0.0228

4111/2 2H9/2 4F5/2

4G5/2

41:;7/2 4G7/2 2K13/2

211u 2 515 5I 6 5F z 3H a 5F 3 5G 2 5G 3 3K 6 30 2 a n 5 3F 2

4563 9631 9921 11 220 11 518 13297 16133 21585 3954 7231 8197 10 752 11 588 11 853 12 732 15087 18424 21809 22595

0.0214 0.0544 0.2016 0.7029 0.0392 0.0884 0.0346 0.0134 0.0192 0.0134 0.2275 0.0215 0.0114 0.4629 0.1493 0.0150 0.0141 0.0184 0.0145

Sm 3+ pu 3+

(6H5/2) 6H7/2 1080 0.2062 (6H5/2) 6H7/2 3327 0.1312

6H9/2 2290 0.0256 6H9/2 6254 0.0541

6F1/2 6397 0.1939 6F3/2 6745 0.1481

6F3/2 6641 0.1444 6F5/2 6751 0.0144

6F5/2 7131 0.0332 6F1/2 6925 0.1513

41)1/2 26573 0.0001 419/2 17471 0.0222

Eu 3+ Am 3+

(TFo) 7F 2 1018 0.1375 (TFo)

5D 2 21499 0.0008

Gd3 + f Cm 3 +

(8S7/2) (8S7/2)

5350 21709 21961

20208 22949 28078 30644 7F 2

5D 2 SG 2 6D5/2

619/2 6D7/2 6D5/2

0.0961 0.0065 e 0.0112 e

0.0108 0.0115 0.0191 0.0137

TABLE 3 (continued)

Transition Transition

L a n t h a n i d e Excited energy Calculated Actinide Excited energy Calculated

ion c state ( c m - 1) U~2) ion c state ( c m - 1) U~z)

Tb a + Bk 3 +

(7F6) 7F 5 2112 0.5376 (7F6)

7F 4 3370 0.0889

VF s 4566 0.1373

7F,, 5156 0.4901

5H~ 21020 0.0121

5H 7 21 036 0.0429

5I 6 26 071 0.0164

5F 4 26 529 0.0155

5L 7 26 858 0.0103

Dy3 + Cf3 +

(6H15/2) 6H13/2 3506 0.2457 (6H15/2) 6Fll/2 6453 0.8901

6Hll/2 5833 0.0923 6H13/z 8103 0.2098

6F11/2 7730 0.9387 2Hll/2 11 585 0.1648

4115/2 22293 0.0073 6H15/2 16 172 0.0258

4Gll/2 23321 0.0004 4K17/2 20869 0.0798

4K17/2 26 365 0.0109 4113/2 23 176 0.0616

4K15/2 276•4 0.0143

4115/2 29 135 0.0176

4Hll/2 29517 0.0398

4H13/2 30306 0.0178

H o 3 + Es 3 +

(518) 517 5116 0.0250 (518)

3K 8 21 308 0.0208

5G 6 22094 1.5201

3H 6 27 675 0.2155

3L 9 29020 0.0185

317 38 470 0.0157

5I 7 11 039 0.0188

5I 6 13 009 0.5686

3K 8 19 904 0.0779

5G 6 19 933 ]1.0614

Er 3 + F m 3 +

(4115/2) 4113/2 6610 0.0195 (4115/2) 2Hl1/2 10037 0.7517

4111/2 9256 0.7125 4111/2 21 365 0.7534

4G11/2 26 496 0.9183

2K15/2 27801 0.0219

T m 3 + M d 3 +

(3H6) 3F 4 5811 0.5375 (3H6) 3F 4 3500 0.5828

3H 5 8390 0.1074 3H 5 15487 0.1048

3H 4 12720 0.2373 3H 4 19088 0.2177

1G 4 21 374 0.0483

lI 6 34886 0.0106

a The principal SLJ c o m p o n e n t of the state is given as a label only. In m a n y cases, the state has < 5 0 ~ of the indicated character.

b Bands experimentally observed to be hypersensitive are included even if the calculated value of U t2) is

<0.01.

c G r o u n d - s t a t e term symbol is shown in parantheses.

o N o calculations reported (Carnall 1989b) for this ion.

It is not k n o w n whether only one or both of these two states is hypersensitive.

eNo b a n d s have U TM > 0.01 and no observed bands are reported (Henrie et al. 1976) to be hypersensitive.

184 J.V. BEITZ

published a vibronic-coupling model for magnetic-dipole intensities of f - f transitions of octahedral complexes of lanthanides. Their model predicts that phonon-assisted (i.e., vibronic) magnetic-dipole lines will have intensities 100-1000 times weaker than pure (i.e., zero phonon) magnetic-dipole lines and phonon-assisted electric-dipole lines. It seems clear that these small intensities justify neglecting vibronic bands in carrying out Judd-Ofelt model analyses of the absorption spectra of aquated lanthanide ions. No vibronic bands built on 5f states of aquated actinide ions have been reported.

Dalam dokumen Edited by: Karl A. Gschneidner, Jr., LeRoy Eyring, G.H. Lander and G.R. Choppin ISBN: 978-0-444-81724-2 (Halaman 185-196)