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The theory of vibronic transitions in rare earth compounds
J. Phya.
Pergamon
Chem.
.%/I&
Vol.
56. No.
8. pp.
10534M2,
1995
copyeght 0 199sEbevkJ scimccLtd Printedin Great Britaia.All ri#bu resend
002236!V(%poo~
0022-X97/95
THE THEORY OF VIBRONIC TRANSITIONS RARE EARTH COMPOUNDS
S9.50
+ 0.00
IN
0. L. MALTA? Departamento de Quimica Geral e Inorglnica, Instituto de Quimica da UNESP, Rua Prof. Francisco Degni, s/n. Araraquara, 14800-980 SP, Brazil (Received 5 July 1994; accepted 15 December
1994)
Abstract-The theory of vibronic transitions in rare earth compounds is re-examined in the light of a more reliable representation for the ligand field Hamiltonian than the crude electrostatic model. General expressions that take into account the relevant contributions from the forced electric dipole and dynamic coupling mechanisms are derived for the vibronic intensity parameters. These include additional terms, from charge and polarizability gradients, which have not been considered in previous work. Emphasis is given to the relative signs of these various contributions. Under certain approximations these expressions may be conveniently written in terms of accessible ligand field parameters. A comparison with experimental values for the compounds Cs,NaEuCl, and LiEuF, is made and satisfactory agreement between theory and experiment is found. A discussion is given on the sensitivity of the calculated intensities to the values of radial integrals, interconfigurational energy differences and ligand field parameters that may be used. Finally, the problem in which a vibronic and an electronic level are in resonance, or near resonance, is analyzed. Suitable expressions to describe the effects of the even-rank components of the vibronic Hamiltonian are obtained. It is found that, depending on the strength of the vibronic interaction and the resonance conditions, the admixture between these two levels may lead to intensities of nearly equal values.
1. INTRODUCMON Vibronic
transitions
in rare earth
compounds
have
been studied on the basis of the theoretical approaches
by Faulkner and Richardson [ 11, Judd [2] and Stavola et al. [3]. These approaches have in common the basic idea in which the small deviation, during a vibrational motion, from the equilibrium symmetry leads to additional ligand field interactions that may be able to admix opposite parity configurations of the rare earth ion, as pointed out by Dexpert-Ghys and Auzel[4]. This is particularly important when the rare earth ion is in a centrosymmetric site. The effect of vibrations on the dynamic coupling (DC) mechanism of 4f-4f intensities [S-8] was considered only in the former two approaches [I, 21. A considerable amount of experimental data on vibronic intensities in rare earth compounds has been obtained. These data have been discussed in a recent review by Blasse [9]. There the author expresses his feeling that the theoretical approaches are so complicated that they are not suitable for a satisfactory 7 On leave of absence from the Departamento de Quimica Fundamental, CCEN, UFPE 50670-901,Recife, PE, Brazil.
rationalization of the experimental data. On this basis most of these data have been analyzed [9-121 in terms of the compact expressions obtained by Judd [2] through the full use of tensor operator techniques. These expressions, however, are rigorously valid for centrosymmetric compounds only. While appreciating the arguments for such a procedure one must keep in mind that the complexity of any theoretical approach on this problem would be hardly overcome due to the inherent complexity of the vibronic interactions. Therefore, in the cases of non-centrosymmetric compounds the procedure adopted by Faulkner and Richardson [l] or a generalized form of Judd’s results [2] would be more consistent. A further problem arises in the computation of theoretical vibronic intensities. It concerns the reliability of the values of quantities such as radial integrals, interconfigurational energy differences, polarizabilities and ligand field parameters. Indeed, this difficulty is already encountered in the calculation of zero-phonon 4f-4f intensities. Differences in the values that may be used for these quantities, when taken together, may lead to an overall effect that may change the order of magnitude of the calculated intensity, as will be discussed later.
1053
0.
1054
L. MALTA
In this paper the theory of rare earth vibronic intensities is re-examined. The theoretical approach takes into account both the forced electric dipole (ED) and DC mechanisms and it is developed in a way that the main contributions to the predicted intensities can be in principle analyzed individually. This point is of great importance since they may have opposite signs. One of our main objectives is to show that under certain approximations these contributions may be conveniently expressed in terms of usual ligand field parameters. As mentioned above, one of the problems in the computation of the vibronic intensities lies in the ligand field interaction. In our approach we use a ligand field model [ 13, 141which has proved to be more satisfactory than the crude electrostatic approximation used in the previous approaches [1,2]. Besides its advantage, from the quantitative point of view, this model takes into account the dependence of the effective charges, which produce the ligand field interaction, on the distances between central ion and ligands. This leads to an additional contribution, due to the gradients of these charges, which has been overlooked in the previous treatments. Analogously a further contribution appears when the ligand polarizabilities are taken to be distance dependent. Finally, we examine the interesting problem which arises when a vibronic level is in resonance, or near resonance, with an electronic level. As we will show, this is a situation which may give useful information about the nature and strength of the vibronic interaction. Further, the expressions obtained are suitable for treating the effects of the even parity components of the vibronic Hamiltonian and might be particularly useful in the analysis of magnetic dipole vibronic transitions.
2. VIBRONIC MATRIX ELEMENTS AND INTENSITY
PARAMETERS
In what follows the magnetic dipole operator is not taken into account. The interaction with the radiation field, a, is written as
In these equations the index i is associated with the electrons of the rare earth ion and j with a ligand at position fij having a polarizability 4; e is the electronic charge (-4.8 x 10-Loe.s.u.). For a given normal mode of vibration, Q, the dipole operator hoc is expanded around the equilibrium position as
_,Q. The ligand field Hamiltonian
(3)
is also expanded
as VI
where CF) is a Racah tensor operator. In the simple overlap model [13, 141the ligand field parameters 7:) are given by
=
C r~Y_i)
where pj is the magnitude of the overlap between the 4f and jth ligand wavefunctions. The factors /?,.are approximately equal to 1. In this model the ligand field is produced by effective charges, gjpje, situated around the middle distances Rj/2. The overlap pi is allowed to be distance dependent through the relation:
0
%”
pJ =po F where, for rare earth ions, it is assumed that p. = 0.05 and R,, is the smallest among the Rjvalues. The power n is assumed to be 3.5 [15]. The vibronic eigenstates are taken beyond the crude Born-Oppenheimer approximation through first-order perturbation theory:
where where In the above equation fine is the dynamic coupling dipole operator, which is given by [7]
(8) $i and x,, representing electronic and vibrational wavefunctions, respectively. We are then interested in evaluating the matrix element:
1055
Vibronic transitions in rare earth compounds
aQ
a qwj) [ R!+’ I
1
= (2k + l)(k + l)“* 1 (- lY+’ q’.m
1
k
X m
a2
G
k
--4
.ftp,=
+
1)
Y$$If!i’~~).
4‘
-y$(_l)-Y$(n,)
m
Jm
+...
(9)
The following considerations are now made. The DC dipole operator in eqn (2) when expanded in terms of irreducible tensor operators contains even and odd components. We have systematically estimated its odd components contribution in comparison with the ED operator and found that it is in general smaller by an order of magnitude. Thus, only its even components will be considered in our analysis. Secondly, as mentioned in the introduction, the effects of the even components of the ligand field Hamiltonian will be conveniently taken into account later in the last section of this paper. Therefore, in eqn (9) only the odd components of V are considered. To evaluate the derivatives with respect to the normal coordinate Q we assume that the reference frame is permanently centered at the rare earth ion. Therefore, we may take
---a aQ
aIi
_
(10)
aQ'v
where
a2
,,=C(-1,”
m
(
g
>m
(11)
h_,
P_, being a unit spherical vector [ 161.In this reference frame the derivative of the center of gravity position with respect to Q no longer vanishes. We give below the expansions used in the development of our expressions [ 16, 171:
and the product: y;)(n,)yy)(D,)
= 1 (- l)K+k+‘+“+q K.M
1
3(2k + 1)(2K + 1) I’* X 47l
x
Ygyi-l,).
(16)
where we have assumed a distance dependence for aj of the Clausius-Mossotti type (aRj). The perturbation sums in eqn (9) may be performed by the use of the procedure given in the standard Judd-Ofelt Theory [ 18, 191or, alternatively, by the use of the average energy denominator method (AEDM) as described in detail in Ref. [14]. In this latter a single average energy denominator, AE, is used, which in the case of rare earth ions, is given by the energy difference between the barycenters of the ground 4f” and excited 4fN-‘5d configurations. Tensor operator techniques are then used to find effective operators acting within the 4f” configuration. It has been shown that the two procedures give very similar results [14]. In the present work we adopted the AEDM method. It is found that the matrix element on the left-hand side of eqn (9) is given by
(r(/fi’IFl$ic>=e C (-11ptqG9+ 1) I.l.PJ/ x (2k + l)]‘”
1
-qU
x
k-l
4’
1
k
X
4
( 4
1 -P-4
r P>
(12) (17)
0. L. MALTA
I056
where the BLIP are the usual intensity parameters which appear in the zero-phonon intensity theory. They, therefore, contribute to the so-called A-process or phonon replica transitions. In eqn (17) we have used the closure relation Z& (x,.]x,~)(x,~]x~) = (x,,]xV). Refore presenting the B$ parameters we proceed with the following assumption. In order to relate the sums over ligands, which appear in eqn (17), with the ligand field parameters $ we use eqn (5) (with jl, = 1). To get more useful expressions, we replace R, in eqns (14) and (15) by an average value R and in the DC contributions we replace uj and gjp,, for identical ligands, by average values u and gp. We give below the expressions for each contribution to the B’$, values: +A+*) b, = -[(A + 2)(21 + 3)(21 + 5)]“2r
&0
1
p’.m.i
x
b
_
(212
+
ffl
(
(A+
2-
$
b,=-+B(i-1,l) Vf
PI >
CYR.
(> 32
(18)
(21 + 1)
1
“2(1_
;
“ol)(:,
“-_d
;)
K.A-2
+
’
Tj 6K,,
(23)
uA)
(; 0> I’*
(
x(ZK+l)(A
m
w + 2)(2J+ 5)
l XC mI A+ P p’,mj
- l,n)
112
I
[
x
fort=A+l,and
b5 = -[A(21 - 1)(21 + l)]1/2g@(i
1+2
P
m
#‘2”(j)
3)
(21)
(>
x8(1+1,1)
xc
b4= -3
lp+2)yg+v(j )(!!>n(19)
(20)
for t=l-1. These equations deserve some comment: 6, , bg, b, and 6, correspond to the ED contribution while b2 and b4 correspond to the DC contribution. We have used the vibrational matrix element (x, IQ 1x0) = (h/2wo)“2, where w. is the angular frequency of the normal mode Q. The factors (1 - al) correspond to the screening of multipolar fields acting on the 4f electrons [2,7]. The first point to be noted is that these contributions may have opposite signs. In the previous paper [14] we have pointed out that the sign of the energy denominators which appear in the perturbation sums involved in the theory of 4f-4f transitions has been incorrectly used (it is actually negative). More recently, another sign mistake has been pointed out in a comment by Xia and Reid [20]. When these two are taken into account it turns out that the ED and DC contributions to the zerophonon intensities have opposite signs, in contrast to the prediction made by eqn (17) of Ref. [ 141.In the development of the present expressions this has been considered.
Vibronic transitions in rare earth compounds
We note that a structure of the type:
is common to all the bi values in eqns (18)-(23). This corresponds to a generalized form of the tensors T, F, G and X introduced by Judd [2]. The contributions bs, b, and b, arise from the distance dependence of p, and a,. We point out that, provided reliable models are used, the values of these latter parameters should be weakly dependent on the way p, and CZ, are allowed to vary with R,. An interesting fact occurs in b,. It may be noted from eqn (23) that for K = I - 2 and 1 = 2 the spherical component of the repulsive part of Madelung’s potential, yb’), plays a role. Its physical meaning is related to the fact that the effective charge gradients may couple with the dipolar component of the vibrating ligand field to give a scalar tensor. In the work by Judd [2] an additional ligand polarization effect produced by the rare earth’s positive charge is considered. Actually this is part of an overall polarization produced in each ligand by the neighboring ions, which has been included in ligand field calculations based on the crude electrostatic approximation [21]. In view of the ligand field model we are using in our approach, the inclusion of this contribution would not be appropriate.
3. NUMERICAL
give the ligand field parameters defined in eqn (5). The obvious implication of this approximation is that the influence of the appropriate vibrational modes is only distinguished by their frequencies. This is particularly more serious in centrosymmetric compounds. In 0, symmetry, for example, since y$) vanishes, no contribution to b,, for i = 2, is predicted, while indeed this is not so [2]. On the other hand, whenever ykg is allowed by symmetry this approximation is expected to be useful, at least to order of magnitude values, as will be shown by our estimates below. We also expect its drawbacks to be attenuated if integrated vibronic intensities are analyzed rather than individual ones. We will return to this point later. To apply our results we have chosen the compounds Cs, NaEuCl,, which is centrosymmetric, and LiEuF, in which the Eu3+ ion occupies a site symmetry close to D,,. Relative integrated vibronic intensities have been measured in both cases for the ‘Do-+ ‘F, transition [IO]. The reference is the zero-phonon magnetic dipole ‘0, + ‘F, transition. The relative intensity is given by
GA A
“lb 02
I0 I
WOI A “I
(26)
ESTIMATES
This can only be achieved through a detailed normal coordinate analysis, which may be of considerable complexity in low symmetry compounds with usually high coordination numbers. Since our intention is to obtain expressions that can be handled and analyzed as much as possible in terms of quantities which can be readily obtained from structural data or spectra, such as ligand field parameters, we felt tempted to proceed with a rather drastic approximation. It consists in assuming that
Iaq_I 4% _- l
(24)
aQ
and
( >m=& aQ
I”b 0 I
where o is the transition frequency, and A is the corresponding coefficient of spontaneous emission. In the present case:
A;;‘: = 4’;;c!;2)3
In order to use the results of the previous section it is necessary to know the derivatives (aR,/dQ),,.
aR,
1057
(25)
where Mj is an effective reduced mass. Thus, for identical ligands, the sums over j in eqns (18)-(23)
j(Rr’b(SDo 11Cl”’ 11’F2)2
(27)
and A, , = 4.67 x 10-46n;w;. ,
(28)
where 2 = n,(ni + 2/3)2, no (assumed to be 1.5 in both compounds) being the index of refraction. If the so-called Huang-Rhys parameter [9] is small the term B,,,,, (x, Ixv) can be neglected in eqn (17) and then nyb = ;
&’ I&l&12.
(29)
Equations (27) and (29) are obtained from the square of eqn (17) followed by a summation over the Stark components of the initial and final states, as in the Judd-Ofeh theory [18, 191.The prime in eqn (29) means that the sum runs also over the normal modes involved. In Table 1 we list the values of the necessary quantities in our estimates. The Cs,NaEuCl, compound has structural parameters very similar to those
0. L. MALTA
1058
Table 1. Numerical values of the quantities used in the calculation of the vibronic intensities. The ligand field parameters y$) are in units of erg/@), and y?, = (- l)p’y$)‘.The reduced mass M is in atomic mass units. he (r”j, h’E, c2 and B(r, A) are the same as used in Ref. [14] LiEuF,
Cs,NaEuCl,
UP ti’
5.35 x 10-12 0 1.74 x 1020 1 x 1020 226 114 [ 80 28.7 2.5 2.58
o. (cm-‘) M a (i’)
1
g
0:: (cm-‘) aSI (cm-‘)
16074 16849
8.2 x 1O-‘2 2697 -0.92 x 10zo - 1.2 x 1020 446 329 199 [ 153 16.8 1 2.24 1 15941 16888
(r2) =2.75 x lo-“cm2; m= -1.2 x 10Jcm-‘; Q(l,2)= -0.17; (r4) = 1.58 x 10-3’cm4; (r2= 0.6; 8(3,2) = 0.345.
of Cs,NaBkCl, [22]. The ionic radii of Eu3+ and Bk3+ are very close. In the latter compound it has been found that the inter-atomic distance Bk-Cl is 2.58 A [23], which is considerably shorter than the typical values for rare earth chlorine compounds (-3 A). This might be one of the reasons for the high value of the Bi ligand field parameter (1384 cm-‘) found experimentally in Cs,NaEuCl, [24]. Another consequence of this short distance is that one may expect a reduction of the polarizability value of the Cl- ion
(typically 3 A’). In our calculations we have assumed a value of 2.5 A’. In the LiEuF, compound there are four F- ions at a distance of 2.217 A and four others at a distance of 2.27 A [25]. Ligand field and zero-phonon intensity parameters are also available [26,27]. For both compounds the experimental ligand field parameters, Bi = (r2)yb2) and Bi = (r4)yy), are quite well reproduced by using eqn (5) with pi = 0.05 and flj = 1. The vibrational frequencies used were v3 = 226cn-‘, v,= 144cm-’ and vs= 80cm-’ in thecase of the Cs,NaEuCl, compound [28]. From the data presented in Ref. [29] it is difficult to assign what vibrational modes have been taken into account in the integrated intensity ratio in the LiEuF, compound. Apparently only a few modes are operative for the ‘Do+‘F2 vibronics in this case. We then decided to take the vibrational frequencies 446 cm-‘, 329 cm-‘, 199 cm-’ and 153 cm-’ [30]. The estimated B$, and intensity ratios are presented in Tables 2 and 3. The first point to be noted is the relative signs of the various contributions. As expected the ED (6,) and DC (b,) contributions have opposite signs. A second point is that b, dominates, particularly in the Cs,NaEuCl, compound. The contributions from charge and polarizability gradients, b, , b4 and be, are found to be of high relevance. They may be comparable to or even more important than the ED b, and b, contributions. According to our approach B& vanishes. This is
Table 2. Vibronic intensity parameters (in units of lo-” cm) and the intensity ratio I$/&_, for the Cs,NaEuCl, compound. The experimental value of the intensity ratio is 0.4 [lo]. The values of bi are given for the 226 cm-’ normal mode. The calculated Qy’b, from eqn (29), takes into account the three normal modes involved and the relation B”b 2.,.-p (coo =
226 cm-‘)
4 b, 5, b, b, b, Total
B”b 2.3.0
0.47 -4 0.11 0.84 -2.58
=
(-
Bvib 2.3.1
0.286 -2.44 0.067 0.5 -1.587
vw~#)’ Bvib 2.3.3
0.356 -3.04 0.084 0.64 -1.96
B”ib 2.18
1 0 -0.057 - 0.057
,yib 2.1.1
0 0.057 0.057
flyb = 0.74 x lO-2o cm2; &$/IS, = 0.38.
Table 3. The same as in Table 2 for the LiEuF, compound. The b, values are given for the 329 cm-’ normal mode. The experimental intensity ratio is 0.1 [IO] (w. = 329 cm-‘) 4
b, b, b, b, b, Total
BWb 2.3.0 -0.27 0.95 -0.054 -0.02 0.6
RI’&= 0.094 x 10-20cm2; @;/I,,
pib2.3.1
-0.165 0.56 -0.033 -0.013 0.35 = 0.07.
,yib2.3.3
-0.46 1.61 -0.11 -0.12 0.92
B”b2.1.0
-0.032 -0.14 -0.17
,,j”b 2.1.1
-0;16 -0.11 -0.13
Vibronic transitions in rare earth compounds a consequence of the approximations assumed in eqns (24) and (25) which lead to the replacement:
Thus, it may be noted from eqns (I@-(23) that, if the ligand field parameters r?), for p’ = 1,2,3, are not allowed by symmetry, no contribution to B& is predicted. When compared with the results of the previous approaches [l-3] our bi values should, therefore, be taken with care. Suppose, in the case of the Cs,NaEuCl, compound, we assume a hypothetical maximum ~a), which would be simply &-6ya)(j). We then find ]bs] = 0.34 x lo-” cm. This corresponds to assuming that the derivatives (aR,/aQ),,, for the axial and equatorial ligands have the same magnitude and opposite signs. In this case a reduction in the sum: dR-
(>
Cd?(j) J i
aQ
m
by a factor of seven occurs, while aR-
(>
CAYA A aQ /
m
remains unaffected. Under this circumstance a reevaluatron of the B$, leads to n;lb = 0.4 x lo-*’ cm’ which can be compared with the figure of 0.14 x’ lo-*“cm* in Table 2. What we wish to illustrate with this comparison is that the approximations of eqns (24) and (25) are expected to be more useful when treating the integrated n;lb intensity parameters rather than the individual B”& ones. However, it should be noted that the relative values of b, and b2, as well as the relative values of 6, and b4, are independent of the sums over ligands. They are, therefore, independent of the approximations made in eqns (24) and (25). In the previous calculations by Faulkner and Richardson [1] and Judd [2] the contribution corresponding to b, in our approach might have been overestimated in our opinion. Possibly one of the reasons is that these authors have used a crude point charge model for the ligand field, which is known to give in general considerably overestimated low rank ligand field parameters, particularly when it is restricted to the first coordination sphere. Thus, for example, it has been usual to take the point charge B: parameters, in Eu+’ compounds, multiplied by a factor of 0.4 in order to bring them closer to the experimental values [21, 311. Actually, this corresponds to the same screening factor (1 - a,), which
1059
appears in the DC contributions above. It is interesting that in the simple overlap model, eqn (5), for the parameters of rank two the factor p(2/?)‘+ ’ is _ 0.4. This correction is clearly more important for the rank-one parameters. Another possible reason lies in the quantities E(t, 2) used by those authors. These correspond to the factors:
2?‘+‘)@(1 AE
’
L)=r(t
’
A)
(30)
that may be extracted from our expressions. If the HartreeFock intercontigurational radial integrals and energy differences of Smentek-Mielczarek (private communication) are used we find a reasonable agreement between B(f, 2) and r(t, I). This is still valid if the radial integrals (r’+’ ) given by Freeman and Desclaux [32] are used in eqn (30). However, when compared with Krupke’s values [33], as used in Refs [1,2], it is found that E(l, 2) is greater than r(1, 2) by a factor of three. It is appropriate here to call attention to the fact that the signs of Krupke’s S(t, 2) should be reversed due to the overlooked negative sign of the energy differences which appear in these quantities as pointed out in Ref. [14]. Krupke’s values for Z(t, A) assume an energy difference (A&,1 of 72,000 cm-’ for the Eu3+ ion [33]. In our opinion this value, obtained from a linear interpolation, might correspond to the lowest states of the 4y5d configuration, but is too low to be taken for its barycenter. It is difficult to estimate to what extent the radial integrals and interconfigurational energy differences are modified by the presence of the ligands. In our opinion it is hard to see how to see how this could change significantly the relative values of b, and b5. With the wb values for the CsrNaEuCl, compound (Table 2) we have also calculated the dipole strengths, S”, for the ‘F,+ ‘D, and ‘F, + ‘0, transitions. These are given by Sd= e2R;‘b(sD, 1)U”‘//‘F,)*.
(31)
For these transitions the squared reduced matrix elements are 8 x 10m4 and 2.5 x 10-3, respectively [34]. We then find S& = 1.36 x 10m4*and S: , = 4.3 x 10-42, in units of e.s.u.2 cm*, which may be compared with the respective values of 1.2 x lo-‘* and 4.2 x 10m4*,calculated by Faulkner and Richardson [1], and with the experimental values of 4.8 x 10m4’ and 5.5 x 10w4*estimated by these same authors. It is worth mentioning that our flyb values would be increased by a factor of -3 if we had used in our calculations the same radial integrals (r’+‘) used in Refs [1, 21.
1060
0. L. MALTA
In compounds in which there is strong light absorption in the visible-near U.V. region one might expect a considerable increase in the ligand polarizabilities as well as a decrease in the screening factors ul [lo]. If the ligand field parameters of higher ranks are of large values, then we expect the vibronic intensities to be dominated by b,. In this case we may predict, from eqn (19), that n:b should be proportional to
Then it may be shown that the new modified eigenstates are given by [35]
Iw=c~s(~)I~I~~=o) + sin(+r)]42, v = 1) (34) and lti2> = -sin(+)l4,,
0 = 0) +cos(~)(~,,
a2(1 - a,)21
]y~f2’]?
This might explain some abnormally high vibronic intensities for the Eu3+ ion in SrTi03 for example [lo]. Important contributions may also arise from B,,,, (x,, I x0) when the Huang-Rhys factor, S, assume high values, corresponding to the strong vibronic coupling limit. Thus, for example, if in the LiEuF, compound S is as high as 0.01, then (x,]K~) x0.1, and from the values of the Bl,,$ given in Ref. [14] we would expect a contribution of comparable magnitude with respect to the BFF# given in Table 2.
4. RESONANCE BETWEEN VIBRONIC ELECTRONIC LEVELS
AND
We now consider a case in which a vibronic state 1ti2, v = 1) is in resonance, or near resonance, with a zero-phonon non-degenerate state 14,) v = 0). This is shown schematically in Fig. 1. The vibronic interaction is written as
(32)
cut,d+lV’lhn, v>.
2 vi2 tanr=v;,-v;2+d
(33)
I 4!!’
6 =E$‘-(ET)+c)
$1-.-.-•--.-.-.---._
-
I
AE
BE = (S2 + 4V;;)“2.
(38)
The intensity of a transition from a zero-phonon state Id’, v = 0) to the vibronic state I&) has then a contribution proportional to the square of the matrix element: (I(I2IFl4’9
v = 0)
-sin(ff)(&,v
=O(jiI~$‘,v =0)
(39)
where fi is the sum of the forced electric dipole, the dynamic coupling and the magnetic dipole operators. We may, therefore, write (40)
where &,, is the zero-phonon intensity of the transition to the pure electronic state id,, v = 0). The transition to the ate I$,) obviously acquires a vibronic character and its intensity is given by 1;‘b = cos2(L 7 )l 2 tp’
Fig. 1. Energy level scheme showing the resonance, or near resonance, between a vibronic (-y -) and an electronic (-) level. The line (- . -. - .) indicates the energy of the electronic level in the absence of resonance.
(37)
where E\‘) and E$‘) are the zero-phonon energies corresponding to the states Iq$, v = 0) and Iti2, v = 0) and c is the phonon energy of the normal mode Q. With the Hamiltonian of eqn (32) it may be readily seen that Vi, = Vi, = 0. Then, the energy separation, AE, of the two states in question is given by [35]
lq’b= sin2( t r )I,
\
(36)
and
=
where now k assumes even values. We define
a-
(35)
where
P’
Gn =
v = 1)
(41)
It may be noted that the sum of the two intensities is always equal to &; 1:” decreases rapidly as T decreases, and becomes much smaller than Iv for 12Vi2/S 14 1. This corresponds to either a very weak
Vibronic transitions in rare earth compounds
1061
in rare earth compounds, giving emphasis to the relative signs of the mechanisms involved, in the light of a more reliable representation of the ligand field Hamiltonian. The second one was to arrive at theoretical expressions that could be analyzed in terms of accessible ligand field parameters. This was possible by the use of certain approximations in which the influence of the appropriate normal modes are distinguished only by their frequencies. The additional contributions due to the charge and polarizability gradients, which have not been considered in previous approaches [l-3], were shown to be of considerable importance according to our numerical estimates. It was also found that, at least in the present cases, the DC mechanism dominates. The numerical estimates have also indicated that the t = 1 + 2 component of the ED mechanism dominates over the t = I - 1 one. Quite satisfactory agreement was found between theory and experiment. We have also discussed the sensitivity of the calculated intensities on the available values of radial integrals and interconfigurational energy differences, as well as on the model assumed for the ligand field Hamiltonian. Thus, depending on the values taken for these quantities the calculated intensities may differ by more than an order of magnitude. Provided that the Huang-Rhys factor is known the contribution from the A-process, or phonon replicas, could be readily taken into account through the terms BA,,#(x0, Ix,) in eqn (17). There is indeed evidence that corroborates the importance of these terms [38, 391. A more systematic application of the present approach to compounds where structural data and ligand field parameters are available is under preparation. Finally, we have discussed the problem in which a vibronic level is in resonance, or near resonance, with an electronic level. This involves the even-rank components of the vibronic ligand field Hamiltonian. It was concluded that, depending on the strength of the vibronic coupling and the degree of resonance, the two levels may become so mixed as to lead to nearly equal intensities. intensities
I- AE--r Fig. 2. Schematic representation of the effect caused by the resonance between a vibronic and an electronic level on the transition intensities. The dashed curve corresponds to the transition to the electronic level in the absence of resonance.
vibronic coupling or a situation in which the two levels are completely out of resonance. On the other hand, for )2V;,/S I> 1, the two intensities may be of nearly equal values. This is represented schematically in Fig, 2. This effect has been indeed predicted by Brawer [36] in a more general treatment of the interaction between electronic levels and a density of phonon states. Within the approach developed in the previous sections we have estimated typical values for Vi2 and found that they lie in the range 0.5-5 cm-‘. Thus, for values of 6 of this order, which would correspond to a perfectly measurable AE, this effect could be observed. Ideal systems to look for such an effect would be low symmetry rare earth compounds in which two Stark levels belonging to non-degenerate irreducible representations are well isolated and resonance with a particular vibrational mode occurs. A measurement of lTb and lfb would then provide valuable information on the vibronic coupling. If the electronic state IQ,, u = 0) is degenerate further splittings may be predicted. This has been treated in the case of the CeF, compound by Gerlinger and Schaack [37] in terms of phenomenological vibronic coupling constants. A detailed discussion here on the additional vibronic intensities in this case would take us too far and, therefore, will be left to a forthcoming analysis.
5. CONCLUDING
Acknowledgements-The author acknowledges the Conselho National de Desenvolvimento Cientitico e Tecnol6gico (CNPq) and the Funda9Io de Amparo a Pesquisa do Estado de S&o Paul0 (FAPESP) for financial support. He is grateful for the hospitality received at the Instituto de Quimica da UNESP, Araraquara. Thanks are also due to Professor J. A. Varela and Mr J. Arruda for help with the computational facilities.
REMARKS REFERENCES
in this paper had two main objectives. The first one was to re-examine the relevant contributions to vibronic The theoretical
approach
developed
1. Faulkner T. R. and Richardson F. S., Mol.NIJX 35, 1141 (1978). 2. Judd B. R., Physica Scripta 21, 543 (1980).
1062
0. L. MALTA
3. Stavola M., Isganitis L. and Sceats M. G., .I. Chem. Phys. 74, 4228 (1981). 4. Dexpert-Ghys J. and Auzel F., J. Chem. Phys. 80,4003 (1984). 5. Jorgensen C. K. and Judd B. R., Mol. Phys. 8, 281 (1964). 6. Mason S. F., Peacock R, D. and Stewart B., Chem. Phys. Lert. 29, 149 (1974). 7. Judd B. R., J. Chem. Phys. 70, 4830 (1979). 8. Malta 0. L., Mol. Phys. 42, 65 (1981). 9. Blasse G., innt. Reu. Phys. Chem. 11, 71 (1992). 10. Blasse G., hwg. Chin-r. AC& 267, 33 (1990). 1I. Sytsma J., Schaik van W. and Blasse G., J. Phys. Chem. Solids 52, 419 (1991). 12. Donega C. de Mello, Meijerink A. and Blasse G., J. Phys.: Condens. Malter 4, 8889 (1992). 13. Malta 0. L., Chem. Phys. L&t. 88, 353 (1982). 14. Malta 0. L., Ribeiro S. J. L., Faucher M. and Percher P., J. Phys. Chem. Solids 52, 587 (1991). 15. Jorgensen C. K., Modern Aspects of Ligand Field Theory, p. 151. North-Holland, Amsterdam (1971). 16. Edmonds A. R., Angular Momentum in Quanfum Mechanics. Princeton University Press, Princeton, NJ (1957). 17. Judd B. R., Operator Techniques in Atomic Speclroscopy. McGraw-Hill, New York (1963). 18. Judd B. R., Phys. Rev. 127, 750 (1962). 19. Ofelt G. S., J. Chem. Phys. 37, 511 (1962). 20. Xia S.-D. and Reid M. F., J. Phys. Chem. Solidr 54,777 (1993). 21. Faucher M. and Garcia D., Phys. Rev. B 26, 5451 (1982). 22. Morss L. R., Siegal M., Stenger L. and Edeistein N., Inorg. Chem. 9, 1771 (1970).
23. Morss L. R. and Fuger J., Inorg. Chem. 8, 1433 (1969). 24. Serra 0. A and Thompson L. C., Inorg. Chem. 15, SO4 (1976). 25. Thoma R. E., Bunton G. D., Peenneman R. A. and Keenan T. K., Inorg. Gem. 9, 1096 (1970). 26. Ghrller-Walrand C., Behets M., Percher P., MouneMinn 0. K. and Laursen I., Inorg. C&m. Acta 109, 83 (1985). 27. Giirller-Walrand C., Behets M., Percher P. and Carnal1 W. T., J. Less-Common Met. 126, 271 (1986). 28. Choca M., Ferraro J. R. and Nakamoto K., Coord. Chem. Rev. 12, 295 (1974).
29. van Vliet J. P. M., van der Voort D. and Blasse G., J. Luminesc. 42, 305 (1989).
30. Miller S. A., Rast H. E. and Caspers H. H., J. Chem. Phys. 52, 4172 (1970). 31. Percher P., Thdse de Doctorat d’Etat, Univ. Pierre et Marie Curie--Paris 6 (1977). 32. Freeman A. J. and Desclaux J. P., J. Magn. Magn. Mater. 12, il (1979). 33. Krupke W. F., Phys. Reo. 145, 325 (1966). 34 Carnal W. T., Crosswhite H. and Crosswhite H. M., Energy Level Structure and Transition Probabilities of the Trivalent Lanthanides in LOF,. Argonne National Laboratory Report (1977). 35 Davydov A. S., Quantum Mechanics. Pergamon Press, Oxford (1965). 36 Brawer S., J. Chem. Phys. 68, 3352 (1978). 37 Gerlinger H. and Schaack G., Phys. Rev. B 33, 7438 (1986).
38 Auzel F., de Sri G. F. and Azevedo W. M., J. Luminesc. 21, 187 (1980). 39 Donega C. de M. and Meijerink A., J. Luminesc. 55,315 (1993).
Pergamon
Chem.
.%/I&
Vol.
56. No.
8. pp.
10534M2,
1995
copyeght 0 199sEbevkJ scimccLtd Printedin Great Britaia.All ri#bu resend
002236!V(%poo~
0022-X97/95
THE THEORY OF VIBRONIC TRANSITIONS RARE EARTH COMPOUNDS
S9.50
+ 0.00
IN
0. L. MALTA? Departamento de Quimica Geral e Inorglnica, Instituto de Quimica da UNESP, Rua Prof. Francisco Degni, s/n. Araraquara, 14800-980 SP, Brazil (Received 5 July 1994; accepted 15 December
1994)
Abstract-The theory of vibronic transitions in rare earth compounds is re-examined in the light of a more reliable representation for the ligand field Hamiltonian than the crude electrostatic model. General expressions that take into account the relevant contributions from the forced electric dipole and dynamic coupling mechanisms are derived for the vibronic intensity parameters. These include additional terms, from charge and polarizability gradients, which have not been considered in previous work. Emphasis is given to the relative signs of these various contributions. Under certain approximations these expressions may be conveniently written in terms of accessible ligand field parameters. A comparison with experimental values for the compounds Cs,NaEuCl, and LiEuF, is made and satisfactory agreement between theory and experiment is found. A discussion is given on the sensitivity of the calculated intensities to the values of radial integrals, interconfigurational energy differences and ligand field parameters that may be used. Finally, the problem in which a vibronic and an electronic level are in resonance, or near resonance, is analyzed. Suitable expressions to describe the effects of the even-rank components of the vibronic Hamiltonian are obtained. It is found that, depending on the strength of the vibronic interaction and the resonance conditions, the admixture between these two levels may lead to intensities of nearly equal values.
1. INTRODUCMON Vibronic
transitions
in rare earth
compounds
have
been studied on the basis of the theoretical approaches
by Faulkner and Richardson [ 11, Judd [2] and Stavola et al. [3]. These approaches have in common the basic idea in which the small deviation, during a vibrational motion, from the equilibrium symmetry leads to additional ligand field interactions that may be able to admix opposite parity configurations of the rare earth ion, as pointed out by Dexpert-Ghys and Auzel[4]. This is particularly important when the rare earth ion is in a centrosymmetric site. The effect of vibrations on the dynamic coupling (DC) mechanism of 4f-4f intensities [S-8] was considered only in the former two approaches [I, 21. A considerable amount of experimental data on vibronic intensities in rare earth compounds has been obtained. These data have been discussed in a recent review by Blasse [9]. There the author expresses his feeling that the theoretical approaches are so complicated that they are not suitable for a satisfactory 7 On leave of absence from the Departamento de Quimica Fundamental, CCEN, UFPE 50670-901,Recife, PE, Brazil.
rationalization of the experimental data. On this basis most of these data have been analyzed [9-121 in terms of the compact expressions obtained by Judd [2] through the full use of tensor operator techniques. These expressions, however, are rigorously valid for centrosymmetric compounds only. While appreciating the arguments for such a procedure one must keep in mind that the complexity of any theoretical approach on this problem would be hardly overcome due to the inherent complexity of the vibronic interactions. Therefore, in the cases of non-centrosymmetric compounds the procedure adopted by Faulkner and Richardson [l] or a generalized form of Judd’s results [2] would be more consistent. A further problem arises in the computation of theoretical vibronic intensities. It concerns the reliability of the values of quantities such as radial integrals, interconfigurational energy differences, polarizabilities and ligand field parameters. Indeed, this difficulty is already encountered in the calculation of zero-phonon 4f-4f intensities. Differences in the values that may be used for these quantities, when taken together, may lead to an overall effect that may change the order of magnitude of the calculated intensity, as will be discussed later.
1053
0.
1054
L. MALTA
In this paper the theory of rare earth vibronic intensities is re-examined. The theoretical approach takes into account both the forced electric dipole (ED) and DC mechanisms and it is developed in a way that the main contributions to the predicted intensities can be in principle analyzed individually. This point is of great importance since they may have opposite signs. One of our main objectives is to show that under certain approximations these contributions may be conveniently expressed in terms of usual ligand field parameters. As mentioned above, one of the problems in the computation of the vibronic intensities lies in the ligand field interaction. In our approach we use a ligand field model [ 13, 141which has proved to be more satisfactory than the crude electrostatic approximation used in the previous approaches [1,2]. Besides its advantage, from the quantitative point of view, this model takes into account the dependence of the effective charges, which produce the ligand field interaction, on the distances between central ion and ligands. This leads to an additional contribution, due to the gradients of these charges, which has been overlooked in the previous treatments. Analogously a further contribution appears when the ligand polarizabilities are taken to be distance dependent. Finally, we examine the interesting problem which arises when a vibronic level is in resonance, or near resonance, with an electronic level. As we will show, this is a situation which may give useful information about the nature and strength of the vibronic interaction. Further, the expressions obtained are suitable for treating the effects of the even parity components of the vibronic Hamiltonian and might be particularly useful in the analysis of magnetic dipole vibronic transitions.
2. VIBRONIC MATRIX ELEMENTS AND INTENSITY
PARAMETERS
In what follows the magnetic dipole operator is not taken into account. The interaction with the radiation field, a, is written as
In these equations the index i is associated with the electrons of the rare earth ion and j with a ligand at position fij having a polarizability 4; e is the electronic charge (-4.8 x 10-Loe.s.u.). For a given normal mode of vibration, Q, the dipole operator hoc is expanded around the equilibrium position as
_,Q. The ligand field Hamiltonian
(3)
is also expanded
as VI
where CF) is a Racah tensor operator. In the simple overlap model [13, 141the ligand field parameters 7:) are given by
=
C r~Y_i)
where pj is the magnitude of the overlap between the 4f and jth ligand wavefunctions. The factors /?,.are approximately equal to 1. In this model the ligand field is produced by effective charges, gjpje, situated around the middle distances Rj/2. The overlap pi is allowed to be distance dependent through the relation:
0
%”
pJ =po F where, for rare earth ions, it is assumed that p. = 0.05 and R,, is the smallest among the Rjvalues. The power n is assumed to be 3.5 [15]. The vibronic eigenstates are taken beyond the crude Born-Oppenheimer approximation through first-order perturbation theory:
where where In the above equation fine is the dynamic coupling dipole operator, which is given by [7]
(8) $i and x,, representing electronic and vibrational wavefunctions, respectively. We are then interested in evaluating the matrix element:
1055
Vibronic transitions in rare earth compounds
aQ
a qwj) [ R!+’ I
1
= (2k + l)(k + l)“* 1 (- lY+’ q’.m
1
k
X m
a2
G
k
--4
.ftp,=
+
1)
Y$$If!i’~~).
4‘
-y$(_l)-Y$(n,)
m
Jm
+...
(9)
The following considerations are now made. The DC dipole operator in eqn (2) when expanded in terms of irreducible tensor operators contains even and odd components. We have systematically estimated its odd components contribution in comparison with the ED operator and found that it is in general smaller by an order of magnitude. Thus, only its even components will be considered in our analysis. Secondly, as mentioned in the introduction, the effects of the even components of the ligand field Hamiltonian will be conveniently taken into account later in the last section of this paper. Therefore, in eqn (9) only the odd components of V are considered. To evaluate the derivatives with respect to the normal coordinate Q we assume that the reference frame is permanently centered at the rare earth ion. Therefore, we may take
---a aQ
aIi
_
(10)
aQ'v
where
a2
,,=C(-1,”
m
(
g
>m
(11)
h_,
P_, being a unit spherical vector [ 161.In this reference frame the derivative of the center of gravity position with respect to Q no longer vanishes. We give below the expansions used in the development of our expressions [ 16, 171:
and the product: y;)(n,)yy)(D,)
= 1 (- l)K+k+‘+“+q K.M
1
3(2k + 1)(2K + 1) I’* X 47l
x
Ygyi-l,).
(16)
where we have assumed a distance dependence for aj of the Clausius-Mossotti type (aRj). The perturbation sums in eqn (9) may be performed by the use of the procedure given in the standard Judd-Ofelt Theory [ 18, 191or, alternatively, by the use of the average energy denominator method (AEDM) as described in detail in Ref. [14]. In this latter a single average energy denominator, AE, is used, which in the case of rare earth ions, is given by the energy difference between the barycenters of the ground 4f” and excited 4fN-‘5d configurations. Tensor operator techniques are then used to find effective operators acting within the 4f” configuration. It has been shown that the two procedures give very similar results [14]. In the present work we adopted the AEDM method. It is found that the matrix element on the left-hand side of eqn (9) is given by
(r(/fi’IFl$ic>=e C (-11ptqG9+ 1) I.l.PJ/ x (2k + l)]‘”
1
-qU
x
k-l
4’
1
k
X
4
( 4
1 -P-4
r P>
(12) (17)
0. L. MALTA
I056
where the BLIP are the usual intensity parameters which appear in the zero-phonon intensity theory. They, therefore, contribute to the so-called A-process or phonon replica transitions. In eqn (17) we have used the closure relation Z& (x,.]x,~)(x,~]x~) = (x,,]xV). Refore presenting the B$ parameters we proceed with the following assumption. In order to relate the sums over ligands, which appear in eqn (17), with the ligand field parameters $ we use eqn (5) (with jl, = 1). To get more useful expressions, we replace R, in eqns (14) and (15) by an average value R and in the DC contributions we replace uj and gjp,, for identical ligands, by average values u and gp. We give below the expressions for each contribution to the B’$, values: +A+*) b, = -[(A + 2)(21 + 3)(21 + 5)]“2r
&0
1
p’.m.i
x
b
_
(212
+
ffl
(
(A+
2-
$
b,=-+B(i-1,l) Vf
PI >
CYR.
(> 32
(18)
(21 + 1)
1
“2(1_
;
“ol)(:,
“-_d
;)
K.A-2
+
’
Tj 6K,,
(23)
uA)
(
x(ZK+l)(A
m
w + 2)(2J+ 5)
l XC mI A+ P p’,mj
- l,n)
112
I
[
x
fort=A+l,and
b5 = -[A(21 - 1)(21 + l)]1/2g@(i
1+2
P
m
#‘2”(j)
3)
(21)
(>
x8(1+1,1)
xc
b4= -3
lp+2)yg+v(j )(!!>n(19)
(20)
for t=l-1. These equations deserve some comment: 6, , bg, b, and 6, correspond to the ED contribution while b2 and b4 correspond to the DC contribution. We have used the vibrational matrix element (x, IQ 1x0) = (h/2wo)“2, where w. is the angular frequency of the normal mode Q. The factors (1 - al) correspond to the screening of multipolar fields acting on the 4f electrons [2,7]. The first point to be noted is that these contributions may have opposite signs. In the previous paper [14] we have pointed out that the sign of the energy denominators which appear in the perturbation sums involved in the theory of 4f-4f transitions has been incorrectly used (it is actually negative). More recently, another sign mistake has been pointed out in a comment by Xia and Reid [20]. When these two are taken into account it turns out that the ED and DC contributions to the zerophonon intensities have opposite signs, in contrast to the prediction made by eqn (17) of Ref. [ 141.In the development of the present expressions this has been considered.
Vibronic transitions in rare earth compounds
We note that a structure of the type:
is common to all the bi values in eqns (18)-(23). This corresponds to a generalized form of the tensors T, F, G and X introduced by Judd [2]. The contributions bs, b, and b, arise from the distance dependence of p, and a,. We point out that, provided reliable models are used, the values of these latter parameters should be weakly dependent on the way p, and CZ, are allowed to vary with R,. An interesting fact occurs in b,. It may be noted from eqn (23) that for K = I - 2 and 1 = 2 the spherical component of the repulsive part of Madelung’s potential, yb’), plays a role. Its physical meaning is related to the fact that the effective charge gradients may couple with the dipolar component of the vibrating ligand field to give a scalar tensor. In the work by Judd [2] an additional ligand polarization effect produced by the rare earth’s positive charge is considered. Actually this is part of an overall polarization produced in each ligand by the neighboring ions, which has been included in ligand field calculations based on the crude electrostatic approximation [21]. In view of the ligand field model we are using in our approach, the inclusion of this contribution would not be appropriate.
3. NUMERICAL
give the ligand field parameters defined in eqn (5). The obvious implication of this approximation is that the influence of the appropriate vibrational modes is only distinguished by their frequencies. This is particularly more serious in centrosymmetric compounds. In 0, symmetry, for example, since y$) vanishes, no contribution to b,, for i = 2, is predicted, while indeed this is not so [2]. On the other hand, whenever ykg is allowed by symmetry this approximation is expected to be useful, at least to order of magnitude values, as will be shown by our estimates below. We also expect its drawbacks to be attenuated if integrated vibronic intensities are analyzed rather than individual ones. We will return to this point later. To apply our results we have chosen the compounds Cs, NaEuCl,, which is centrosymmetric, and LiEuF, in which the Eu3+ ion occupies a site symmetry close to D,,. Relative integrated vibronic intensities have been measured in both cases for the ‘Do-+ ‘F, transition [IO]. The reference is the zero-phonon magnetic dipole ‘0, + ‘F, transition. The relative intensity is given by
GA A
“lb 02
I0 I
WOI A “I
(26)
ESTIMATES
This can only be achieved through a detailed normal coordinate analysis, which may be of considerable complexity in low symmetry compounds with usually high coordination numbers. Since our intention is to obtain expressions that can be handled and analyzed as much as possible in terms of quantities which can be readily obtained from structural data or spectra, such as ligand field parameters, we felt tempted to proceed with a rather drastic approximation. It consists in assuming that
Iaq_I 4% _- l
(24)
aQ
and
( >m=& aQ
I”b 0 I
where o is the transition frequency, and A is the corresponding coefficient of spontaneous emission. In the present case:
A;;‘: = 4’;;c!;2)3
In order to use the results of the previous section it is necessary to know the derivatives (aR,/dQ),,.
aR,
1057
(25)
where Mj is an effective reduced mass. Thus, for identical ligands, the sums over j in eqns (18)-(23)
j(Rr’b(SDo 11Cl”’ 11’F2)2
(27)
and A, , = 4.67 x 10-46n;w;. ,
(28)
where 2 = n,(ni + 2/3)2, no (assumed to be 1.5 in both compounds) being the index of refraction. If the so-called Huang-Rhys parameter [9] is small the term B,,,,, (x, Ixv) can be neglected in eqn (17) and then nyb = ;
&’ I&l&12.
(29)
Equations (27) and (29) are obtained from the square of eqn (17) followed by a summation over the Stark components of the initial and final states, as in the Judd-Ofeh theory [18, 191.The prime in eqn (29) means that the sum runs also over the normal modes involved. In Table 1 we list the values of the necessary quantities in our estimates. The Cs,NaEuCl, compound has structural parameters very similar to those
0. L. MALTA
1058
Table 1. Numerical values of the quantities used in the calculation of the vibronic intensities. The ligand field parameters y$) are in units of erg/@), and y?, = (- l)p’y$)‘.The reduced mass M is in atomic mass units. he (r”j, h’E, c2 and B(r, A) are the same as used in Ref. [14] LiEuF,
Cs,NaEuCl,
UP ti’
5.35 x 10-12 0 1.74 x 1020 1 x 1020 226 114 [ 80 28.7 2.5 2.58
o. (cm-‘) M a (i’)
1
g
0:: (cm-‘) aSI (cm-‘)
16074 16849
8.2 x 1O-‘2 2697 -0.92 x 10zo - 1.2 x 1020 446 329 199 [ 153 16.8 1 2.24 1 15941 16888
(r2) =2.75 x lo-“cm2; m= -1.2 x 10Jcm-‘; Q(l,2)= -0.17; (r4) = 1.58 x 10-3’cm4; (r2= 0.6; 8(3,2) = 0.345.
of Cs,NaBkCl, [22]. The ionic radii of Eu3+ and Bk3+ are very close. In the latter compound it has been found that the inter-atomic distance Bk-Cl is 2.58 A [23], which is considerably shorter than the typical values for rare earth chlorine compounds (-3 A). This might be one of the reasons for the high value of the Bi ligand field parameter (1384 cm-‘) found experimentally in Cs,NaEuCl, [24]. Another consequence of this short distance is that one may expect a reduction of the polarizability value of the Cl- ion
(typically 3 A’). In our calculations we have assumed a value of 2.5 A’. In the LiEuF, compound there are four F- ions at a distance of 2.217 A and four others at a distance of 2.27 A [25]. Ligand field and zero-phonon intensity parameters are also available [26,27]. For both compounds the experimental ligand field parameters, Bi = (r2)yb2) and Bi = (r4)yy), are quite well reproduced by using eqn (5) with pi = 0.05 and flj = 1. The vibrational frequencies used were v3 = 226cn-‘, v,= 144cm-’ and vs= 80cm-’ in thecase of the Cs,NaEuCl, compound [28]. From the data presented in Ref. [29] it is difficult to assign what vibrational modes have been taken into account in the integrated intensity ratio in the LiEuF, compound. Apparently only a few modes are operative for the ‘Do+‘F2 vibronics in this case. We then decided to take the vibrational frequencies 446 cm-‘, 329 cm-‘, 199 cm-’ and 153 cm-’ [30]. The estimated B$, and intensity ratios are presented in Tables 2 and 3. The first point to be noted is the relative signs of the various contributions. As expected the ED (6,) and DC (b,) contributions have opposite signs. A second point is that b, dominates, particularly in the Cs,NaEuCl, compound. The contributions from charge and polarizability gradients, b, , b4 and be, are found to be of high relevance. They may be comparable to or even more important than the ED b, and b, contributions. According to our approach B& vanishes. This is
Table 2. Vibronic intensity parameters (in units of lo-” cm) and the intensity ratio I$/&_, for the Cs,NaEuCl, compound. The experimental value of the intensity ratio is 0.4 [lo]. The values of bi are given for the 226 cm-’ normal mode. The calculated Qy’b, from eqn (29), takes into account the three normal modes involved and the relation B”b 2.,.-p (coo =
226 cm-‘)
4 b, 5, b, b, b, Total
B”b 2.3.0
0.47 -4 0.11 0.84 -2.58
=
(-
Bvib 2.3.1
0.286 -2.44 0.067 0.5 -1.587
vw~#)’ Bvib 2.3.3
0.356 -3.04 0.084 0.64 -1.96
B”ib 2.18
1 0 -0.057 - 0.057
,yib 2.1.1
0 0.057 0.057
flyb = 0.74 x lO-2o cm2; &$/IS, = 0.38.
Table 3. The same as in Table 2 for the LiEuF, compound. The b, values are given for the 329 cm-’ normal mode. The experimental intensity ratio is 0.1 [IO] (w. = 329 cm-‘) 4
b, b, b, b, b, Total
BWb 2.3.0 -0.27 0.95 -0.054 -0.02 0.6
RI’&= 0.094 x 10-20cm2; @;/I,,
pib2.3.1
-0.165 0.56 -0.033 -0.013 0.35 = 0.07.
,yib2.3.3
-0.46 1.61 -0.11 -0.12 0.92
B”b2.1.0
-0.032 -0.14 -0.17
,,j”b 2.1.1
-0;16 -0.11 -0.13
Vibronic transitions in rare earth compounds a consequence of the approximations assumed in eqns (24) and (25) which lead to the replacement:
Thus, it may be noted from eqns (I@-(23) that, if the ligand field parameters r?), for p’ = 1,2,3, are not allowed by symmetry, no contribution to B& is predicted. When compared with the results of the previous approaches [l-3] our bi values should, therefore, be taken with care. Suppose, in the case of the Cs,NaEuCl, compound, we assume a hypothetical maximum ~a), which would be simply &-6ya)(j). We then find ]bs] = 0.34 x lo-” cm. This corresponds to assuming that the derivatives (aR,/aQ),,, for the axial and equatorial ligands have the same magnitude and opposite signs. In this case a reduction in the sum: dR-
(>
Cd?(j) J i
aQ
m
by a factor of seven occurs, while aR-
(>
CAYA A aQ /
m
remains unaffected. Under this circumstance a reevaluatron of the B$, leads to n;lb = 0.4 x lo-*’ cm’ which can be compared with the figure of 0.14 x’ lo-*“cm* in Table 2. What we wish to illustrate with this comparison is that the approximations of eqns (24) and (25) are expected to be more useful when treating the integrated n;lb intensity parameters rather than the individual B”& ones. However, it should be noted that the relative values of b, and b2, as well as the relative values of 6, and b4, are independent of the sums over ligands. They are, therefore, independent of the approximations made in eqns (24) and (25). In the previous calculations by Faulkner and Richardson [1] and Judd [2] the contribution corresponding to b, in our approach might have been overestimated in our opinion. Possibly one of the reasons is that these authors have used a crude point charge model for the ligand field, which is known to give in general considerably overestimated low rank ligand field parameters, particularly when it is restricted to the first coordination sphere. Thus, for example, it has been usual to take the point charge B: parameters, in Eu+’ compounds, multiplied by a factor of 0.4 in order to bring them closer to the experimental values [21, 311. Actually, this corresponds to the same screening factor (1 - a,), which
1059
appears in the DC contributions above. It is interesting that in the simple overlap model, eqn (5), for the parameters of rank two the factor p(2/?)‘+ ’ is _ 0.4. This correction is clearly more important for the rank-one parameters. Another possible reason lies in the quantities E(t, 2) used by those authors. These correspond to the factors:
2?‘+‘)@(1 AE
’
L)=r(t
’
A)
(30)
that may be extracted from our expressions. If the HartreeFock intercontigurational radial integrals and energy differences of Smentek-Mielczarek (private communication) are used we find a reasonable agreement between B(f, 2) and r(t, I). This is still valid if the radial integrals (r’+’ ) given by Freeman and Desclaux [32] are used in eqn (30). However, when compared with Krupke’s values [33], as used in Refs [1,2], it is found that E(l, 2) is greater than r(1, 2) by a factor of three. It is appropriate here to call attention to the fact that the signs of Krupke’s S(t, 2) should be reversed due to the overlooked negative sign of the energy differences which appear in these quantities as pointed out in Ref. [14]. Krupke’s values for Z(t, A) assume an energy difference (A&,1 of 72,000 cm-’ for the Eu3+ ion [33]. In our opinion this value, obtained from a linear interpolation, might correspond to the lowest states of the 4y5d configuration, but is too low to be taken for its barycenter. It is difficult to estimate to what extent the radial integrals and interconfigurational energy differences are modified by the presence of the ligands. In our opinion it is hard to see how to see how this could change significantly the relative values of b, and b5. With the wb values for the CsrNaEuCl, compound (Table 2) we have also calculated the dipole strengths, S”, for the ‘F,+ ‘D, and ‘F, + ‘0, transitions. These are given by Sd= e2R;‘b(sD, 1)U”‘//‘F,)*.
(31)
For these transitions the squared reduced matrix elements are 8 x 10m4 and 2.5 x 10-3, respectively [34]. We then find S& = 1.36 x 10m4*and S: , = 4.3 x 10-42, in units of e.s.u.2 cm*, which may be compared with the respective values of 1.2 x lo-‘* and 4.2 x 10m4*,calculated by Faulkner and Richardson [1], and with the experimental values of 4.8 x 10m4’ and 5.5 x 10w4*estimated by these same authors. It is worth mentioning that our flyb values would be increased by a factor of -3 if we had used in our calculations the same radial integrals (r’+‘) used in Refs [1, 21.
1060
0. L. MALTA
In compounds in which there is strong light absorption in the visible-near U.V. region one might expect a considerable increase in the ligand polarizabilities as well as a decrease in the screening factors ul [lo]. If the ligand field parameters of higher ranks are of large values, then we expect the vibronic intensities to be dominated by b,. In this case we may predict, from eqn (19), that n:b should be proportional to
Then it may be shown that the new modified eigenstates are given by [35]
Iw=c~s(~)I~I~~=o) + sin(+r)]42, v = 1) (34) and lti2> = -sin(+)l4,,
0 = 0) +cos(~)(~,,
a2(1 - a,)21
]y~f2’]?
This might explain some abnormally high vibronic intensities for the Eu3+ ion in SrTi03 for example [lo]. Important contributions may also arise from B,,,, (x,, I x0) when the Huang-Rhys factor, S, assume high values, corresponding to the strong vibronic coupling limit. Thus, for example, if in the LiEuF, compound S is as high as 0.01, then (x,]K~) x0.1, and from the values of the Bl,,$ given in Ref. [14] we would expect a contribution of comparable magnitude with respect to the BFF# given in Table 2.
4. RESONANCE BETWEEN VIBRONIC ELECTRONIC LEVELS
AND
We now consider a case in which a vibronic state 1ti2, v = 1) is in resonance, or near resonance, with a zero-phonon non-degenerate state 14,) v = 0). This is shown schematically in Fig. 1. The vibronic interaction is written as
(32)
cut,d+lV’lhn, v>.
2 vi2 tanr=v;,-v;2+d
(33)
I 4!!’
6 =E$‘-(ET)+c)
$1-.-.-•--.-.-.---._
-
I
AE
BE = (S2 + 4V;;)“2.
(38)
The intensity of a transition from a zero-phonon state Id’, v = 0) to the vibronic state I&) has then a contribution proportional to the square of the matrix element: (I(I2IFl4’9
v = 0)
-sin(ff)(&,v
=O(jiI~$‘,v =0)
(39)
where fi is the sum of the forced electric dipole, the dynamic coupling and the magnetic dipole operators. We may, therefore, write (40)
where &,, is the zero-phonon intensity of the transition to the pure electronic state id,, v = 0). The transition to the ate I$,) obviously acquires a vibronic character and its intensity is given by 1;‘b = cos2(L 7 )l 2 tp’
Fig. 1. Energy level scheme showing the resonance, or near resonance, between a vibronic (-y -) and an electronic (-) level. The line (- . -. - .) indicates the energy of the electronic level in the absence of resonance.
(37)
where E\‘) and E$‘) are the zero-phonon energies corresponding to the states Iq$, v = 0) and Iti2, v = 0) and c is the phonon energy of the normal mode Q. With the Hamiltonian of eqn (32) it may be readily seen that Vi, = Vi, = 0. Then, the energy separation, AE, of the two states in question is given by [35]
lq’b= sin2( t r )I,
\
(36)
and
=
where now k assumes even values. We define
a-
(35)
where
P’
Gn =
v = 1)
(41)
It may be noted that the sum of the two intensities is always equal to &; 1:” decreases rapidly as T decreases, and becomes much smaller than Iv for 12Vi2/S 14 1. This corresponds to either a very weak
Vibronic transitions in rare earth compounds
1061
in rare earth compounds, giving emphasis to the relative signs of the mechanisms involved, in the light of a more reliable representation of the ligand field Hamiltonian. The second one was to arrive at theoretical expressions that could be analyzed in terms of accessible ligand field parameters. This was possible by the use of certain approximations in which the influence of the appropriate normal modes are distinguished only by their frequencies. The additional contributions due to the charge and polarizability gradients, which have not been considered in previous approaches [l-3], were shown to be of considerable importance according to our numerical estimates. It was also found that, at least in the present cases, the DC mechanism dominates. The numerical estimates have also indicated that the t = 1 + 2 component of the ED mechanism dominates over the t = I - 1 one. Quite satisfactory agreement was found between theory and experiment. We have also discussed the sensitivity of the calculated intensities on the available values of radial integrals and interconfigurational energy differences, as well as on the model assumed for the ligand field Hamiltonian. Thus, depending on the values taken for these quantities the calculated intensities may differ by more than an order of magnitude. Provided that the Huang-Rhys factor is known the contribution from the A-process, or phonon replicas, could be readily taken into account through the terms BA,,#(x0, Ix,) in eqn (17). There is indeed evidence that corroborates the importance of these terms [38, 391. A more systematic application of the present approach to compounds where structural data and ligand field parameters are available is under preparation. Finally, we have discussed the problem in which a vibronic level is in resonance, or near resonance, with an electronic level. This involves the even-rank components of the vibronic ligand field Hamiltonian. It was concluded that, depending on the strength of the vibronic coupling and the degree of resonance, the two levels may become so mixed as to lead to nearly equal intensities. intensities
I- AE--r Fig. 2. Schematic representation of the effect caused by the resonance between a vibronic and an electronic level on the transition intensities. The dashed curve corresponds to the transition to the electronic level in the absence of resonance.
vibronic coupling or a situation in which the two levels are completely out of resonance. On the other hand, for )2V;,/S I> 1, the two intensities may be of nearly equal values. This is represented schematically in Fig, 2. This effect has been indeed predicted by Brawer [36] in a more general treatment of the interaction between electronic levels and a density of phonon states. Within the approach developed in the previous sections we have estimated typical values for Vi2 and found that they lie in the range 0.5-5 cm-‘. Thus, for values of 6 of this order, which would correspond to a perfectly measurable AE, this effect could be observed. Ideal systems to look for such an effect would be low symmetry rare earth compounds in which two Stark levels belonging to non-degenerate irreducible representations are well isolated and resonance with a particular vibrational mode occurs. A measurement of lTb and lfb would then provide valuable information on the vibronic coupling. If the electronic state IQ,, u = 0) is degenerate further splittings may be predicted. This has been treated in the case of the CeF, compound by Gerlinger and Schaack [37] in terms of phenomenological vibronic coupling constants. A detailed discussion here on the additional vibronic intensities in this case would take us too far and, therefore, will be left to a forthcoming analysis.
5. CONCLUDING
Acknowledgements-The author acknowledges the Conselho National de Desenvolvimento Cientitico e Tecnol6gico (CNPq) and the Funda9Io de Amparo a Pesquisa do Estado de S&o Paul0 (FAPESP) for financial support. He is grateful for the hospitality received at the Instituto de Quimica da UNESP, Araraquara. Thanks are also due to Professor J. A. Varela and Mr J. Arruda for help with the computational facilities.
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approach
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