2~/' (CvK~)
R. MARCHAND
3. Definition of terms employed in intensity theory
SPECTRAL INTENSITIES OF f-f TRANSITIONS 109 For orthorhombic, monoclinic and triclinic crystal fields, the labels a, o and 7c cannot be used. Here the nomenclature is to say that an ED transition is allowed in x, y or z polarization and an MD transition is allowed in Rx, Rv or Rz polarization.
An effective method for the prediction of the magnetic dipole character of a transition is intensity calculation. The intensity of a magnetic dipole transition can be calculated if appropriate wavefunctions are available (see sect. 4). Wavefunctions are obtained from a set a free-ion (and crystal-field) parameters. The parameter sets are derived from the energetic positions of the transitions. If a zero or nearly zero intensity is calculated for the magnetic dipole contribution of a particular transition observed in the spectrum, we can conclude that this transition has mainly an induced electric dipole character.
I l0 C. GI3RLLER-WALRAND AND K. BINNEMANS
band (/?max). The /? values can be calculated from the absorbance A by use of Lambert- Beer's law:
A = e • C . d [dim:/], (1)
where e is the molar absorptivity t [dim: L 2, units: mol-~ lcm-1], C is the concentration [dim: L -3, units: mol 1-1], and d is the optical pathlength [dim: L, units: cm],
Although the ema× values can be given also for lanthanide compounds, this practice is not very accurate, especially not for lanthanide ions in a crystalline host matrix, since the ratio of the spectral bandwidth to the natural bandwidth is often larger than 0.1, even with the smallest slit width of the spectrophotometer. When the spectral bandwidth is one-tenth of the natural bandwidth of the peak, deviation from the true peak height is less than 0.5%.
A larger spectral bandwidth will increase this error. A smaller spectral bandwidth will increase the noise without significantly improving the quality of the data. If the absorption band is not completely resolved, the observed bandwidth is wider than the natural bandwidth and the absorption maximum is lower (emax is reduced). This is the reason why, upon the availability of high resolution recording spectrophotometers, researchers found higher emax values for the same lanthanide systems than their predecessors with low-resolution equipment. The maximum molar absorptivity of an absorption band can be used for the spectrophotometric determination of trivalent lanthanide ions, because in this case calibration curves have to be constructed. The fact that an absorption band is not totally resolved is not a problem in this case, as long as the same slit width is used for recording the spectra of both the standards and the samples. A procedure for the spectrophotometric dosage of lanthanide ions has been given by Banks and Klingman (1956) and by Stewart and Kato (1958).
It is not always possible to know the exact doping concentration of lanthanide ions diluted in a host crystal or in a glass. Often, the nominal concentration of the components is taken, i.e. the concentration of the starting composition of the batch. However, the true concentration may deviate from this value, because of losses of volatile compounds and because of inhomogeneities in the matrix. The concentration C of lanthanide ions doped into a host single crystal is given by
1027
C - x Z x 6
Vu×NA [dim: L -3, units: moll-l], (2)
where V~ is the volume of the crystallographic unit cell [dim: L 3, units: ~3], NA is Avogadro's number [dim:/], Z is the number of formula units per unit cell [dim:/], and 6 is the doping fraction (e.g. 6 = 0.05 for a doping concentration of 5%) [dim:/].
As opposed to doped matrices, for undiluted lanthanide salts the concentration can be so high that the linear relationship between absorbance and concentration no longer holds (violation of Beer's Law). If the two crystal faces through which the light beam
In older textbooks, the term molar extinction coefficient is used instead of molar absorptivity.
SPECTRAL INTENSITIES OF f-f TRANSITIONS l 1 l travels are not parallel or if the crystal faces are not fiat, there will be a variation in the optical pathlength. The determination of the pathlength of tiny crystals is also a factor of uncertainty. Reflection losses may result from defects and inclusions inside the crystal. The reflection losses will not cause major problems, because these result in a gradual increase of the baseline towards the ultraviolet. The peaks of the lanthanide ions are simply superposed on this broad band absorption, in the case of single crystals and glasses, often no reference sample is placed in the reference beam. For solutions, the reference sample is the solvent. The reader can find further information about apparent deviations from the law of Lambert-Beer in textbooks on analytical chemistry.
The area under an absorption peak is a better measure of the intensity than the molar absorptivity at the peak maximum, because the area is the same for both the resolved and the unresolved band. The area can be determined by integrating the peak, which is equivalent to the calculation of the integral
/
e(~) d~ [dim: L], (3)where F is the wavenumber [dim:
L l,
units: cmr].
The integrated molar absorptivity can be seen as the sum of the e(F) values over the whole absorption band. Instead of the molar absorptivity, the dipole strength D or the oscillator strength P are reported. One has to be cautious of the fact that the sample can be an oriented crystal or a medium in which the molecules are randomly oriented (solution, glass, powder). Throughout the text, we will emphasize this orientational problem. We use the symbols D r and U for oriented systems, and D and P for randomly-oriented systems.
3.3. Dipole strength (D r) and oscillator strength (P~) of a single spectral line in an oriented system
A single spectral line in an oriented system is equivalent to the case of a single crystal studied with polarized light. Here, the molar absorptivity e(V) at a wavenumber ~ is related to the dipole strength by:
- 8Jr3 23~3 hc _ _ _ f(-9) ~D' [dim: L2], (4)
where
NA = 6.02214 × 1023 (Avogadro's number) h = 6.62554× 10 -27 ergs (Planck's constant) c = 2.997925 × 10 j° cm s -j (speed of light)
~: wavenumber
f ( F ) : line shape function Dr: dipole strength
[dim:/], [dim: M L 2 T-I], [dim: L T-I], [dim: L-l], [dim: L],
[dim: M L 5 T -2 or e 2 L2].
112 C. GORLLER-WALRAND AND K, BINNEMANS
The line shape function must fulfil the following conditions:
(a) f f ( - ~ ) V n dV = V~ [dim: L-"] (5)
and f f ( v ) dV = 1 [dim:/], (6)
Of(V)
(b) ~ dV = 0 [dim: L], (7)
[ of(v) v "v =
(c)
a 1 [dim:/]. (8)The dipole strength D' is defined as the absolute square of the matrix element in the dipole operator 0(p 1) (MD or ED operator) between the wavefunction tPi of the initial state and the wave function of ~ f of the final state:
Dr= (I/.z 0; 1) (iuf) 2= (t/./i {);1)t/./f)* ((,U i ~;1)t/If)
= 0(1) 2
p,f-i [dim: M L 5 T -2 or e2L 2, units: esu2 cm2],(9)
where O(p ~) corresponds to a single component of the transition electric or magnetic dipole
( /"
in the molecular axis system ( p can be x , y , z or +1,-1,0). The term gti O(pl) qtf is the complex conjugate of ( ~i O~ 1) tpf). For instance, the dipole strength of an induced electric dipole transition of an oriented molecule with its x-axis parallel to the electric field vector is given by:
D' : I(V' I, ?t V' )I 2
[dim: e 2 L2], (10)where
m(x 1) is the electric dipole operator.The integrated molar absorptivity is
g(V) 8s~ 3
NA
D ' l f ( V ) V d V f [dim: L], (11)d ~ =
hc 2303 J or, by using eq. (5),
r e ( v ) dV = 8Jr3 NA ¥0D' [dim: L], (12)
hc 2303
with v0 the wavenumber of the absorption maximum (in cm -1). Filling in the constants in eq. (12) results in
f 8 x (3. 14159) 3 x (6.02214 x 1023) -9oD' , (13)
e(V) dV = (6.62554 x 10-27ergs) × (2.997925 × 101°cms -1) x 2303
SPECTRAL INTENSITIES OF f - f TRANSITIONS 113
o r
J e(V) dV = 3 × 108.9 × 1036 V0 D l [dim: L; D' in esu 2 cm2]. ( 14) In eq. (14), the dipole strength D ~ is expressed in esu 2 cm 2. Often, D / is expressed in Debye2 (D2). The conversion is straightforward, since 1 Debye = 10 -18 esucm:
re(V)
d ~ = 3 × 108.9VoD ~ [dim: L; D ' in Debye 2 (D2)]. (15) It is also possible to report D / in cm 2, by multiplying the right-hand side o f eq. (14) by e 2:/
e(V) dV = 3 × 108.9 × 1036yoe2D ~
= 0.75 × 102°VoD ~ [dim: L; D ' in cm2].(16) In eq. (12) it is assumed that the absorption band is symmetric (i.e., the band can be described by a Gaussian or Lorentzian line shape function). It is however more general to determine the integral
f[e(V)/V]
d r :/ e ( V ) d V -
~ -
8J'83hc
~ NAD r /
f ( V ) dV [dim: L 2] (17)o r
e(v)
-=-_ dV = 3 × 108.9 × 1036D /
v (18)
[dim: L 2, units: tool -1 lcm-1; D ~ in esu 2 cm2].
The dipole strength D p is related to the oscillator strength U by
_ 8 ~ 2 m e c ~ ( l ) 2
P~ 8¢C2meC v 0 D t - Vo [dim:/], (19)
he 2 he 2 p,
f-iwhere
e = 4.803 x 10-1°esu (elementary charge), me = 9.10904 × 10 -28 g (electron mass),
h = 6.6261 × 10 -27 ergs (Planck's constant), c = 2.997925 × 10 I° cm s -~ (speed o f light),
Vo = wavenumber at the absorption m a x i m u m (in cm-l).
Upon filling in the constants, eq. (19) results in
8 × (3.14159) 2 x (9.10904 × 10-18g) x (2.997925 × 1 0 1 ° c m s - l ) ¥ o D ~ p , =
(6.6261 × 10-27ergs)(4.803 × 10-1°esu) (20)
= 4.702 × 10 -7 × 3 × 1036 × ~o × D~
114 C. GORLLER-WALRAND AND K. BINNEMANS
pl = 1.41 x 1030 x Vo × D 1, (21)
and
D' = (7.089 x 10 -31 x U ) [D' in esu 2 cm2]. (22)
~0
Equations (21) and (22) can be used to convert the dipole strength D ~ to the oscillator strength P~ and vice versa (in oriented systems).
Since
2303he f e(V) dV [dim: e 2 L], (23)
Vo D~- 82g3N A
eq. (19) can be rewritten as /~ = 8yrZmec 2303hc f e(V) d~
he 2 8;V3NA
J
_ 2303 mec 2 f e(~) dV(24) NAZr e 2
J
2303 × (9.109o4 × 10-2~g) × (2.997925 × 101°cms-~) 2 /
d~, (6.o2214 × 1023) × 3.14159 × (4.g03 × 10-~°esu) ~
or
P' = 4.32 x 10 -9 f e(~) dV [dim:/]. (25)
Equation (25) relates the oscillator strength pr to the integrated molar absorptivity f E(V) dr.
As defined here, the dipole strength D r (eq. 9) and the oscillator strength U (eq. 19) are theoretical quantities. In sect. 3.6 we will introduce the experimental dipole strength Drexp and oscillator strength Uex p, which can be compared directly with the calculated values.
3.4. Dipole strength (D) and oscillator strength (P) of a transition in a randomly- oriented system
Randomly-oriented systems are ions in solutions, glasses or powders. These systems are studied by unpolarized light. In this case, the molar absorptivity e(-9) at a wavenumber
is related to the dipole strength D by 8Jr 3 _ NA 1
e(V) = ~ c f(-9) v 2 - - ~ ~t) [dim: L2], (26)
where