4.1. Magnetic dipole matrix element for a single spectral line in an oriented system The calculated magnetic dipole strength for a transition between a state with wavefunction (F v r SLJM I and a state with wavefunction [l N r/S'L I J I M l) can be found by evaluating the matrix element in the dipole operator OMD. This operator is also denoted as Op(1). Pay attention to the fact that several authors use this notation for the electric dipole operator.
The magnetic dipole matrix element is given by
[dim: M 1/2 L 5/2 T -1 or eL, units: esu cm],
(60) where i. is the total orbital angular momentum operator, S is the total spin angular momentum operator, h = h/2ar, and eh/(2mec) = 9.273 x 10 -21 esucm. However, note that instead of L + 2,~, one should strictly write I. + geS, where ge is the electron g-factor (Atkins 1988) and its numerical value is 2.00223. This approximation has little effect on the calculated magnetic dipole strengths. The magnetic dipole operator/~p{1) is a tensor
122 C• GORLLER-WALRAND AND K. B1NNEMANS
of rank 1 with components p. The components p are called polarization numbers. For a magnetic dipole transition in the o-polarization p = 0, for a and Jr polarization p = ±1:
o ( p = 0) • r,(1)_ ~(I)
:~0 - ~z
-2mec eh (£z ~.~,(1)
+ z & , ) 0 , ( 6 1 )c~,z~ ( p ±1)" ~(1) 1
eh
(£+2~)i~) (62)= .±, = T - ~ ( ~ ) ± i~(y l') -
2meC 5:1"The magnetic dipole operator O(pl) has the intrinsic Cooh symmetry. The calculation of the magnetic dipole matrix elements has been discussed by Shortley (1940) and by Pasternak (1940). The wavefunctions used for the calculation of the magnetic dipole strength are pure 4f N eigenfunctions of the even components (k = even) of the crystal- field Hamilton±an. The wavefunctions are obtained using the same parameter set (both free ion and crystal-field parameters) required for the energy level calculation. It is assumed that the incident light does not interact with the ligand charge distributions. The matrix element in eq. (60) is calculated by application of the Wigner-Eckart theorem (Weissbluth 1978) to remove the p dependence,
1 j , ( 6 3 )
Here, P M/ is a 3-j symbol and its value is tabulated in the work of Rotenberg et al. (1959). The reduced matrix element
(INrSLJ
( L + 2 S ) (1)INr'stL'J ')
can be worked out by splitting the terms in L and S and by an additional application of the Wigner-Eekart theorem (Weissbluth 1978). For the matrix element in L one finds
(l N r SLJ ]lzll l N r'S'L'J')
{ L J S}[(2L+I)(2J+I)(2J,+I)L(L+I)]I/2
=6Tr, SSS, 6LZ,(--1)
s+r+J+I j , L 1(64) where the Kronecker delta 6/f = 0 if i ~ f and 6/f = 1 if i = f .
For the matrix element in S, the result is analogous:
(1N ~ SLJ IIsII 1N r' StL/J r )
=6rr'6SS'dLL'(--1)S+L+J+I { S' JS L } [(2S+ I)(2J + I)(2J' + I)S(S+
(65)
SPECTRAL INTENSITIES OF f-f TRANSITIONS 123 4.2. Selection rules for magnetic dipole transitions
eqs. (63)-(65), the selection rules for magnetic dipole transitions can be derived to F r o m
be
(1)
(2) (3) (4) (5)
A~- = 0, AS = O, AL = O,
AJ = O, ± 1, but 0 ~-+ 0 is forbidden, - M + p + M ' = O, where p = O, • 1,
M ~ - M = 0 (~J polarization), M' - M = i 1 (ct or ~ polarization).
The selection rules on AS and AL are not very strict and are only valid in the Russell- Saunders coupling scheme. In the Russell-Saunders coupling scheme, magnetic dipole transitions will be limited to the J-levels within a single term 2S+~L (i.e. the ground term for transitions observed in the absorption spectrum). Most of these transitions occur at low transition energies (several hundred cm -I) and are thus outside the spectroscopic range for absorption and luminescence measurements. These selection rules can be relaxed in the intermediate coupling scheme, where only J remains a good quantum number. The selection rule on J is more rigid and can only be broken down by J-mixing, which is a weak effect due to the crystal-field perturbation. Because the magnetic dipole operator
~;1) has even parity, magnetic dipole transitions are allowed within one configuration (e.g.
intra-configurational 4 f - 4 f transitions).
With the knowledge of the selection rule AJ = 0, +1, three different cases can be distinguished for the magnetic dipole matrix elements:
(1) J = S ,
IINTSLJ [1(£ + 2S)(1){I INTSLJ} = g [J(J + 1)(2J + 1)] 1/2 , (66) where g is the Land~ factor, given by
J ( J + 1 ) - L ( L + 1 ) + S ( S + 1)
g = 1 + (67)
2 J ( J + 1)
The Landb factor describes the effective magnetic momentum of an atom or electron, in which the orbital angular momentum L and the spin angular momentum S are combined to give a total angular momentum J.
(2) j i = j _ i,
( f i r S L J I1([. + 2,~')(')II lN r S L J - 1}
(68)
= + L + J + 1 ) ( S + L + J - 1 ) ( J + S - L ) ( J + L - S ) , (3) J ' = J + 1,
{IN SLJ II(Z + 2 )<1 111N SLJ + 1}
(69)
[ ~ ( s + L + j 1 iv2
= + 2 ) ( S + J + 1 - L ) ( L + J + 1 - S ) ( S + L - J )
124 C. GORLLER-WALRAND AND K. BINNEMANS
These equations are valid in the Russell-Saunders coupling scheme, for terms which are described by only one quantum number r. In general, a 2S+~Lj Russell-Saunders term is a linear combination of
2S+lLj('r)
terms, where the summation runs over the quantum number r. In the intermediate coupling scheme, for each wavefunction the proper linear combination of the 2S+lLj Russell-Saunders terms with the same J value has to be made before evaluating the matrix elements. The matrix element of a magnetic dipole transition in the intermediate coupling scheme or after J-mixing can therefore be written as(70) where hrsL and r hr,s, c, are the coefficients for the wavefunction in the Russell-Saunders coupling scheme, and aJM and aj,M, are the coefficients for the wavefunction in the intermediate coupling scheme or after J-mixing.
Special note: After evaluation of the magnetic dipole matrix element and taking the absolute square, the calculated magnetic dipole strength is found, expressed in (Bohr magneton) 2 or/32. The matrix element can also be expressed in esu 2 cm 2, by applying the conversion
(71) This transformation is important for mixed ED-MD transitions, where both ED and MD contributions have to be summed (see eqs. 55 and 56).
The expression for the matrix element in eq. (60) is only valid for one spectral line in an oriented system, studied by polarized light. The magnetic dipole strength of a randomly- oriented system is given by
(72) and the oscillator strength for a randomly-oriented system is given by
(73) Sometimes, the symbol Stud (line strength) is used instead of DMD. In practice, the magnetic dipole strength DMI~ for a randomly-oriented system can be obtained by the formula
(74) where the matrix elements in Russell-Saunders coupling scheme are calculated according to eqs. (66)-(69). For the intermediate coupling scheme, eq. (70) has to be considered. As
SPECTRAL INTENSITIES OF f-f TRANSITIONS Table 1
Principal magnetic dipole transitions in the absorption spectra of the trivalent lanthanide ions a 125
Ion Transition Energy ( c m ~ ) b Oscillator strength Dipole strength
(I0 ~)c (10 6 DebyeZ)
Pr 3+ 3H 5 +-- 3H 4 2300 9.76 90
Nd3+ 4Ii i/2 ~-- 419/2 2000 14.11 15
Pm 3+ sI 5 +-- 514 1600 16.36 217
Sm 3+ 6H7/2 +-- 6H5/2 t 100 17.51 339
4G5/2 *-- 6H5/2 17900 1.76 2.1
EH 3- 5Dr) +-- 7F t 16900 7.47 ax 9.4 d'e
5D I ~-- 7F 0 19000 1.62 a 1.8 d
5F t ~- 7F I 33400 2.16 d 1.4 a
Gd3+ ('Pv2 +- 8S7/z 32200 4.I3 2.7
6P5/2 +--- 8S7/2 32 800 2.33 1.5
Tb 3+ 7F 5 ~-- 7F 6 2100 12.11 I23
706 +-- 7176 26400 5.03 4.1
5G 5 +-- 7F 6 27800 6~66 4.9
Dy 3+ 6H13/2 +- 6H~s/2 3500 22.68 138
4115/2 ~ 6HI5/2 22300 5.95 5.7
Mo 3+ 5I 7 +-- 5I 8 5100 29.47 t23
3K s ~ Sl s 21 300 6.39 6.4
Er 3+ 4113/2 +-- 4Ii5/2 6600 30.82 99
2K.Is/2 +~ 4115/2 27 800 3.69 2.8
Tm 3. aH 5 ~-- 3H 6 8400 27.25 69
Yb 3- 2F7/2 *-- 2F5/2 10400 17.76 36
Adapted from Carnall (1979).
b An approximated value of the energy in cm -~ is given (rounded to the nearest hundred).
~ The values are those for transitions in aqueous HCtO 4 solution.
d The population of the 7F 0 (ca. 65%) and 7F~ (ca. 35%) levels at room temperature is taken into account.
Values from G6rller-Walrand et al. (1991).
d e s c r i b e d in sect. 3.6, DMD a n d PMD h a v e to b e m u l t i p l i e d b y Z M J ( 2 J + 1) t o c o m p a r e t h e m w i t h t h e e x p e r i m e n t a l values. I n t a b l e 1, t h e p r i n c i p a l m a g n e t i c d i p o l e ( M D ) t r a n s i t i o n s in t h e a b s o r p t i o n s p e c t r a o f t r i v a l e n t l a n t h a n i d e i o n s a n d t h e i r i n t e n s i t i e s are s u m m a r i z e d ( a f t e r C a r n a l l 1979). T h e v a l u e s are v a l i d for r a n d o m l y - o r i e n t e d s y s t e m s . M a g n e t i c d i p o l e t r a n s i t i o n s are w e a k . T h e i r i n t e n s i t y is in g e n e r a l a f a c t o r o f m a g n i t u d e w e a k e r t h a n e l e c t r i c d i p o l e t r a n s i t i o n s . S e v e r a l t r a n s i t i o n s s h o w a m i x e d d i p o l e c h a r a c t e r , w h i c h m e a n s t h a t t h e y g a i n i n t e n s i t y b y b o t h t h e M D a n d t h e E D m e c h a n i s m s .
4.3. S u m r u l e
T h e i n t e n s i t y o f a n M D t r a n s i t i o n is r e l a t i v e l y i n d e p e n d e n t o f t h e s u r r o u n d i n g s o f t h e l a n t h a n i d e . So, t h e k i n d o f l i g a n d s o r t h e s y m m e t r y o f t h e c o o r d i n a t i o n p o l y h e d r o n a r o u n d
126 C. GORLLER-WALRAND AND K. BINNEMANS
the lanthanide ion do not influence the MD intensity largely. It has been shown by G6rller- Walrand et al. (1991) that in the intermediate coupling scheme, the total magnetic dipole strength of all the crystal-field transitions (over all the polarizations) between two 2S+lLj manifolds is symmetry independent. This sum rule is not valid in the case of J-mixing, but since the J-mixing effect is small for most systems, deviations from the ideal sum rule are expected to be small also. Mean theoretical sum values of 18 x 10 -v D 2, 94 x I 0 -7 D 2 and 9 x 10 -v D 2 have been found for the respective magnetic dipole transitions 5Dl +- 7F0, 5D0 ~ 7F1 and 5D2 +--- 7F1 of Eu 3+. The detailed theoretical derivation of this sum rule is not given here, but can be found in the article by G6rller-Walrand et al. (1991). The influence of J-mixing is especially reflected by the intensity ratio between two crystal- field transitions within one J-manifold (Porcher and Caro 1980).