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Mechanisms of non-radiative energy transfer involving lanthanide ions revisited

O.L. Malta

Departamento de Qumica Fundamental, Universidade Federal de Pernambuco-CCEN, Cidade Universitria, Recife-PE 50670-901, Brazil

a r t i c l e i n f o

Article history:

Available online 5 September 2008

PACS:

31.70.Hq33.70.Ca

Keywords:

Molecular orbitalLuminescenceOptical spectroscopyUp-conversionRare-earths in glasses

a b s t r a c t

In this work, the theoretical background of the Coulomb direct and exchange interactions in non-radia-tive energy transfer involving lanthanide ions is critically reviewed. Emphasis is given to the shieldingeffect produced by the filled 5s and 5p sub-shells of the lanthanide ion and the appropriate use of theso-called forced electric dipole intensity parameters of 4f4f transitions, two aspects that have been oftenoverlooked in the literature. An effective exchange hamiltonian, based on Mlliken-type approximations,and an expression for the resonant or quasi-resonant energy mismatch spectral overlap factor are pro-posed. Both cases of ion-to-ion and intramolecular energy transfer processes are discussed. Numericalestimates are compared with previous calculations, showing that order of magnitude differences appear.The analysis indicates that the donoracceptor distance dependence of transfer rates by the exchangemechanism goes beyond the single exponential behavior adopted from Dexters approach.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Non-radiative energy transfer in the nanoscopic scale is a pro-cess of paramount importance to a great number of relevant phe-nomena in nature. A class of particularly interesting systems inwhich this process occurs consists of coordination compounds oflanthanide ions, usually trivalent europium and terbium ions, withorganic ligands. The characteristic lanthanide ion luminescence(4f4f luminescence) in these compounds is a consequence of thestrong absorption in the UV region by one or more ligands, whichthen transfer energy non-radiatively to the lanthanide ion thatsubsequently emits in the visible (down-conversion of light). Inthis way, the emitting 4f level is in general populated much more

efficiently than by direct excitation of the lanthanide ion excitedlevels. Thus, the 4f4f luminescence intensity is the result of a bal-ance between strong absorption by the ligands, ligandlanthanideion energy transfer rates, non-radiative decays and radiative emis-sion rates involved. This phenomenon was first elucidated byWeissman in 1942 [1]. More recently, models describing the mech-anisms of organic ligandlanthanide ion energy transfer and 4f4femission quantum yields have been discussed in the literature [2].

In the case of inorganic materials (crystals and glasses) contain-ing lanthanide ions, particularly noticeable was the discovery ofsequential energy transfer between lanthanide ions, convertinglow energy photons into higher energy ones (up-conversion oflight), by Auzel in 1966 [3,4]. This discovery allowed the compre-

hension of several non-linear processes in doped solid state struc-tures and induced the prediction of new effects in these materials.The widely used up-conversion process has been the subject of in-tense scientific investigation and also technological applicationsthat goes from the production of laser materials to the modelingof optical markers [58].

Even though the fundamentals of non-radiative energy transferinvolving lanthanide ions are well established there are several as-pects not yet firmly understood. The difficulties in dealing withfactors such as the distribution of ions and statistical aspects ofmacroscopic energy transfer (concentration effects in the case ofionion energy transfer), coupled wavefunctions, 4f4f transitionintensities and energy overlap integrals are not the only ones. Dis-

tinguishing between the possible mechanisms governing the en-ergy transfer is certainly a major problem. Luminescencequantum yields and the transient behavior of the luminescencein lanthanide ions compounds are strongly dependent on these fac-tors when energy transfer is operative [9,10].

The present paper is focused on the theoretical background ofthe mechanisms of non-radiative energy transfer involving lantha-nide ions, with emphasis on the electric multipolar and exchangemechanisms. The first aim is to call attention to certain aspectsof crucial importance that are usually overlooked in the literaturewhen pairwise transfer rates are evaluated. These aspects (theuse of JuddOfelt intensity parameters and the inclusion of shield-ing factors) might lead to orders of magnitude differences in calcu-lated transfer rates. The second goal is to propose, on the basis of aMlliken-type approximation for two-center one-electron matrixelements [11,12], an effective exchange interaction hamiltonian

0022-3093/$ - see front matter 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.jnoncrysol.2008.04.023

E-mail address: [email protected]

Journal of Non-Crystalline Solids 354 (2008) 47704776

Contents lists available at ScienceDirect

Journal of Non-Crystalline Solids

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j n o n c r y s o l


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that may be used in the independent model systems, withoutneeding to work with antisymmetrized wavefunctions. Withinour purposes, the magnetic dipoledipole mechanism [13,14],which might be of relevance when the selection rule on the totalangular momentum is DJ6 1, particularly in centrosymmetric

compounds, will not be considered. Finally, a formulation, basedon Gaussian band shapes, is made for the energy overlap integralexpressing the energy mismatch conditions in the transfer process.

2. The case of ion-to-ion energy transfer

2.1. The interaction hamiltonian

The electrons in the 4f sub-shell of a lanthanide ion are stronglyshielded by the filled 5s and 5p sub-shells, leading to rather weakinteractions between the ion and its environment. It is also con-ceivable that the Coulomb interaction between two electronicclouds becomes somehow modified by the optically inactive med-ium connecting the donor and acceptor species. For donoraccep-

tor distances where the multipolar interactions dominate (largedistances) such effect is usually taken into account through thedielectric constant of the medium (e) by simply multiplying theCoulomb hamiltonian by 1/e. For distances where the exchangeinteraction is the dominant one (short distances) this effect canbe considered by working with lanthanide ion 4f wavefunctionsappropriately modified by the weak interaction with the chemicalenvironment. For both distance ranges this effect is not expected tochange figures within order of magnitude calculations, and will notbe explicitly taken into account for our purposes.

In contrast, the shielding factors produced by the filled 5s and5p sub-shells may lead to significant changes in calculated transferrates. Thus, they will be explicitly considered. We anticipate thefact that this shielding effect applies when for the donoracceptor

pair antisymmetrized wavefunctions are not used in the evaluationof matrix elements, or in other words, when matrix elements donot depend explicitly on the donoracceptor overlap integrals.Otherwise, the same physical effect risks to be counted twice. Thisis due to the fact that the overlap integrals naturally incorporatethis shielding.

The interaction hamiltonian between the electronic clouds of adonor (D) and an acceptor (A), indicated in Fig. 1, can be separatedin two parts according to:

H HC Hex X

i;j

e2

rijX

i;j

e2

rijPij; 1

where HC represents the direct Coulomb interaction and Hex the ex-change interaction. In this expression, the indexes i and j run over

donor and acceptor electrons, respectively, and Pij is the exchangeoperator, given by [15]:

Pij 12

2~si ~sj; 2

where~si and~sj are the spin operators of electrons i and j.Evaluation of the exchange interaction matrix elements is in

general a task of enormous complexity [15]. The convenience of

using the exchange hamiltonian in the form of the second termon the right-hand side of Eq. (1) depends on the nature of donorand acceptor species and how wavefunctions are treated. Impor-tant aspects may be elegantly clarified by the used of second quan-tization methods together with irreducible tensor operatortechniques. In this procedure the Coulomb interaction betweenthe two electronic clouds is written as [16]:

HXn;g;k;f

an dg n1g2

e2

r12

k1f2

dfak

Xn;g;k;f

an dg n1g2

e2

r12

f1k2

akdf:

3In this expression a+ and a are creation and annihilation opera-

tors, respectively, for electrons of the acceptor ion, while d+ and d

are creation and annihilation operators of electrons of the donorion. In the case of ion-to-ion energy transfer, Greek letters standfor one-electron 4f wavefunctions (spin-orbitals). The first and sec-ond terms on the right-hand side correspond, respectively, to thedirect Coulomb and exchange interactions.

The common practice is to expand the two-electron interaction(e2/r12) in terms of Racahs irreducible spherical tensor operators,C(k)(X), by using the bipolar expansion [17]. This leads to:

e2

r12 e2

Xk1 ;k2

q1 ;q2 ;Q

1k1 2k1 2k2 1!2k1!2k2! 1

2 k1 k2 k1 k2q1 q2 Q

Ck1k2Q XRRk1k21

rk11 C

k1q1

X1 rk22 Ck2q2 X21 rk1 1 rk2:

4In this expansion it has been assumed that r1 and r2 are smaller

than R. The quantity in () is a 3-j symbol and the quantities (1 rk)are the shielding factors (also known as Sternheimer factors [18])due to the filled 5s and 5p sub-shells in both donor and acceptorions. From a comparison between ligand field models, for lantha-nide compounds, as given by the Simple Overlap Model [19] andby the Crude Electrostatic Approximation, it has been argued thatthe shielding factors may be conveniently expressed in terms ofthe radial overlap integral, q, between the 4f sub-shell and the va-lence shell of a ligating atom in the first coordination sphere of thelanthanide ion, by the following relation [20]:

1 rk q2bk1; 5where b is a number very close to 1. A typical value of the overlap qin lanthanide compounds is 0.05. Thus, for instance, for k = 2, 4 and6 (1 rk) from Eq. (5) takes the values 0.4, 1.6 and 6.4, respectively.These values and their trend agree fairly well with theoretical calcu-lations of Sternheimer factors for lanthanide ions. It should be clearthat Eq. (5) should be regarded as a rather observationalrelationship.

2.2. Wavefunctions

In the RussellSaunders coupling the lanthanide free ion wave-functions are taken in the angular momentum representation asj(4fN)uSLJMJi, where the total angular momentum ~J ~L ~S and urepresents additional quantum numbers necessary to unambigu-

ously define the wavefunction. Taking into account the fact thatthe intra-ion Coulomb repulsion is not diagonal in the quantumnumber u and that the spinorbit interaction is not diagonal inthe quantum numbers u, S and L, these latter are no longer good

RH

ir

H

jrH

ijr

H

D

A

Fig. 1. Coordinate systems used to describe the donoracceptor pair interaction.

O.L. Malta / Journal of Non-Crystalline Solids 354 (2008) 47704776 4771


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quantum numbers. As a consequence, after diagonalization of thefree ion hamiltonian the free ion eigenstates are given by linear

combinations of the wavefunctions j(4fN

)uSLJMJi, that is:j4fNwJMJi

Xu;S;L

CuSLj4fNuSLJMJi 6

with the following condition:Xu;S;L

jCuSLj2 1: 7

Eq. (6) expresses the so-called intermediate coupling scheme.The eigenstates given in this scheme are essential to describe thebehavior of lanthanide ions. A good example is the case of transi-tions between multiplets of different multiplicities, which other-wise cannot be described. A typical case is illustrated by thetransitions between the 5DJ and

7FJ multiplets of the Eu3+ ion.

Labeling a multiplet by the usual notation 2S+1LJ is a mere indica-tion of the dominant component in the summation in Eq. (6). Aninteresting and useful aspect is that, since the chemical environ-ment weakly affects the 4f orbitals, for each lanthanide ion theeigenstates in the intermediate coupling scheme are essentiallythe same for different environments. However, the weak non-spherical part of the ligand field hamiltonian is responsible for ef-fects of crucial importance in the spectroscopy of lanthanide ions,as J-level splitting, J-mixing (the quantum number J slightly devi-ates from a good quantum number) and the mixing of electronicconfigurations of opposite parities, relaxing Laportes rule. This lat-ter effect is in the origin of the forced electric dipole contribution tothe 4f4f intensities, as described by the JuddOfelt theory [21,22].

The independent systems model is fairly adequate to the case ofnon-radiative energy transfer between two interacting lanthanide

ions. According to this model the initial and final two-centersstates of the lanthanide ion-pair, jUi and jU0i, respectively, are gi-ven by

jUi jDijAi jDAi 8and

j U0i jDijAi jDAi; 9where the donor and acceptor states, jDi and jAi, respectively, aregiven in the intermediate coupling scheme (Eq. (6)). The asteriskindicates excited states. The situation is schematically shown inFig. 2.

2.3. Matrix elements

It has been agreed that in the expansion in Eq. (4) the relevantcomponents, for lanthanide ions, correspond to k1 and k2 6 2. Theterm for k1 = k2 = 0 is the isotropic contribution. In this case, the

expansion in Eq. (4) reduces to a constant term equal to (e2/R).Thus, the hamiltonian in Eq. (3) takes the following form:

H0 e

2

R

Xn;g;k;f

1 r02an dg dfakdnkdgf SnfSgkan dg akdf

; 10

where the quantities Sij are the overlap integrals between the 4forbitals of donor and acceptor ions. The first term in the sum re-duces to NDNA, where N represents the number of 4f electrons[16]. Note that shielding does not appear in the exchange termsince, as mentioned before, they are naturally incorporated in theoverlap integrals Sij. Using the relation akdf + dfak = 0, Eq. (10) iswritten as

H0 e2

R1 r02NDNA

Xn;g;k;f

SnfSgkan d

g dfak

!: 11

The first term on the right-hand side of the above equation,obviously, will not contribute to the energy transfer process. Itmay be noted that approximating all the Sij integrals to an averagevalue Sis an unreliable assumption, since by using the relation for

4f electronsXa;b

ca cb ffiffiffi7

pU

00 ; 12

where U(K) is a unit tensor operator [16], matrix elements becomezero, once the tensor operator U(0) is a constant. However, the over-lap integrals Sij may be separated as a product of radial and angularparts. Thus, the second term on the right-hand side of Eq. (11) is di-rectly proportional to the square of the radial overlap integralh4fDj4fAi.

For k1 (or k2) = 0 and k2 (or k1) 0 it is not difficult to see fromEq. (3) that the direct Coulomb interaction will contribute withzero matrix elements, while the exchange hamiltonian will con-tribute with a term linear in the overlap integrals. This linearity

is, however, apparent, since the two-center matrix elements ofthe operators with k2 (or k1) 0, for instance hg2jrk2 Ck2 q2 jk2i, some-

how depend on the 4f4f overlap and are also strongly dependenton the distance R. For the sake of illustration, in the case of a di-pole-type two-center matrix element (k2 (or k1) = 1) if one crudelyuses a Mlliken-type approximation [11,12], then:

hg2jrC1q2 jk2i %1

2Rhg2jk2i 13

and the dependence on the 4f4f overlap becomes explicit. Actually,the exchange interaction will depend on structures of the type:

Pk1q1 n; fPk2q2

g; kan dg akdf; 14where the quantities Pk1 q1 and P

k2q2

may be interpreted as the multi-pole moments of the overlap region, formed by the small overlapbetween the donor and acceptor electronic clouds. If order of mag-nitude estimates are envisaged, it is tempting enough to use a Ml-liken-type approximation and write

hDAjHexjDAi % e2

Rh4fDj4fAi2: 15

Transfer rates would then depend on the fourth power of the4f4f radial overlap. Obviously, in this very crude approximation,the important selection rules on the w and J quantum numbersdo not appear. An alternative is to use the following matrix ele-ment for the exchange interaction:

hDAjHexjDAi % h4fDj4fAi2 DA Xi;j1

2 2~si ~sj

e2

rij

DA

* +;

16where the non-antisymmetrized wavefunctions are given by Eqs.(8) and (9).

D

D

A

A

DONOR ACCEPTOR

**

Fig. 2. Schematic representation of donor and acceptor states involved in theenergy transfer process.

4772 O.L. Malta / Journal of Non-Crystalline Solids 354 (2008) 47704776


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This analysis indicates that the distance dependence of the ex-change mechanism actually goes beyond the single exponentialbehavior, as it has been adopted from Dexters formulation [23].

For k1 and/or k2 = 1, in the direct Coulomb hamiltonian, it is nec-essary to invoke the mixing of opposite parity electronic configura-

tions through the odd components of the ligand field hamiltonian,as in the JuddOfelt theory of 4f4f intensities [21,22]. This parityadmixture relaxes Laportes rule and produces matrix elementsthat are competitive with those parity allowed for k1 and k2 = 2.These k1 and k2 combinations lead to the exchange, dipoledipole,dipolequadrupole and quadrupolequadrupole contributions tothe transfer rates. For the dipoledipole and dipolequadrupolecontributions one is involved with JuddOfelt matrix elements ofthe type [21]:

hAjrC1q jAi X

n;

2

E4f EnX

t;p

XK;q

1q2K 1

h4fjrjnihnjrtj4fihfkC1kihkCtkfi t 1 Kp q q

f 1 t f K

ctphAjUKqjAi;

17where (n,) indicates excited configurations with opposite parity tothe 4fN one and with average energies En. The ctp are the so-calledodd rank ligand field parameters that appear in the expression forthe ligand field hamiltonian:

HLFodd Xt;p;i

ctprti C

tp Xi: 18

For lanthanides t= 1, 3, 5 and 7, and K= 2, 4 and 6. In Eq. (17)h4fjrtjni is a radial integral, hfkC(t)ki is a one-electron reduced ma-trix element and {} is a 6-j symbol.

For k1 and/or k2 = 2 the matrix element involved is:

hAjr2C2q jAi h4fjr2j4fihfkC2kfi1JA

MA

JA 2 JAMA q MA

hwAJAkU2kwAJAi;19

where we have used the WignerEckart theorem, with the eigen-states given in the intermediate coupling scheme (Eq. (6)).

From the above matrix elements, as far as J is considered as agood quantum number, the following selection rule is derived:

jJ Jj 6 j 6 J J; 20where j stands for k1, k2 or K.

2.4. Energy transfer rates

In the independent systems model, according to Fermis goldenrule, energy transfer rates between a donor and an acceptor ionmay be expressed as:

WET 2ph

jhDAjHjDAij2F; 21

where the temperature dependent factor F contains a sum overFranckCondon factors and the appropriate energy mismatch con-ditions (donoracceptor spectral overlap).

To evaluate transfer rates in the intermediate coupling scheme,the common practice is to carry out a sum over the M quantumnumbers weighted by the degeneracies of the donor and acceptor

initial states, 1=2JD 1 and 1/(2JA+1). Kushida has then derived

the following expressions, without including shielding, for the di-poledipole (dd), dipolequadrupole (dq) and quadrupolequadrupole (qq) mechanisms of energy transfer [24]:

Wdd 1 rD1 21 rA12JDJA

4p

3h

e4

R6X

K

XDKhwDJDkUKkwDJDi2

X

K

XAKhwAJAkUKkwAJAi2F; 22

Wdq 1 rD1 21 rA22JDJA

2p

h

e4

R8X

K

XDKhwDJDkUKkwDJDi2

hr2i2AhfkC2kfi2 hwAJAkU2kwAJAi2F; 23

Wqq 1 rD2 21 rA22JDJA

28p

5h

e4

R10

hr2i2Dhr2i2AhfkC2kfi4hwDJDkU2kwDJDi2

hwAJAkU2kwAJAi2F: 24In the above equations [J] = 2J+ 1, hr2i = h4fjr2j4fi and the quantitiesXK are the so-called forced electric dipole JuddOfelt parameters,given by

XK X

t;p

N2t;Kjctpj22t 1 ; 25

where

Nt;K X

n;

2

E4f En h4fjrjnihnjrtj4fihfkC1kihkCtkfi f 1

t f K

:

26From Eq. (5) one may readily notice that shielding strongly af-

fects the transfer rates, particularly the dipoledipole one. Anotherpoint to be stressed here is that the XK quantities intervening inEqs. (22) and (23) are exclusively those due to the forced electricdipole mechanism of 4f4f transitions. It is well-known that, when

treated phenomenologically, the intensity parameters automati-cally absorb the important ligand polarizability dependent dy-namic coupling mechanism. Thus, using XK values determinedexperimentally in the above equations would be conceptually erro-neous. This very fact had already been stressed in Ref. [25].

2.5. The spectral overlap factor F

This factor is in general obtained from information on the spec-tral overlap between donor emission and acceptor absorptionbands. This empirical procedure, depending on its accuracy, mayaccount for the detailed characteristics of the 4f states and theircoupling with the phonon density of states. However, estimatesof F from band shape considerations are certainly worthy. In this

procedure either Lorentzian or Gaussian band shapes are used. Inthe latter case, for resonant or quasi-resonant energy transfer,the following expression has been proposed [26]:

F ln 2ffiffiffiffip

p 1h2cDcA

1

hcD

2 1

hcA

2" #ln 2

( )12

exp 14

2D

hcD 2ln2

21

hcA

2 1

hcD

2 ln 2

DhcD

2ln 2

2664

3775; 27

where hc represents band width at half-height and D is the differ-ence between the donor and acceptor transition energies involvedin the transfer process. Eq. (27) has been successfully used in the

case of energy transfer between molecular units of a polymericchain in a hybrid material [27]. In this way, the temperature depen-dence and phonon-assistance to the transfer rates are taken into ac-count in an implicit effective fashion. Recently, a detailed



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theoretical analysis of the factor Fhas been performed for the caseof intramolecular energy transfer involving ligand-to-metal chargetransfer states in luminescence thermal quenching effects [28]. Typ-ical values of F for a lanthanide ion-pair are in the range of 10121013 erg1.

3. The case of ionligand intramolecular energy transfer

In this case the energy transfer process involves donor andacceptor species with completely different characteristics. Despitesthe fact that this problem has been experimentally elucidated inthe 1940s [1], only in the last decade theoretical approaches haveappeared [2]. This gap might be partially attributed to the compu-tational difficulties involved.

In organic ligands, once the spinorbit interaction is very smallthe spin is to a large extent a good quantum number, and themolecular states have well defined multiplicities. They are eithersinglets or triplets, with the ground state being a singlet. Under thiscircumstance it is not difficult to conclude that the isotropic contri-

bution (k1 = k2 = 0) to matrix elements vanishes for both direct andexchange interactions. In the latter case the ligand matrix elements(singletsinglet and singlettriplet) of the spin operator vanish, incontrast to the ion-to-ion case of energy transfer.

The relevant lowest order contributions come from the dipoleterm (k1 = 1) from the ligand side and the multipolar terms fromthe lanthanide electronic cloud, mainly the dipolar and quadrupo-lar ones (k2 = 1 and 2, respectively). The donoracceptor distance,RL (often between 3 and 5 ), is taken as the distance from the lan-thanide nucleus to the baricenter of the ligand electronic state in-volved in the energy transfer process. Thus, the followingcombinations have been considered [2]: k1 = 1 and k2 = 1, 2, 4and 6, for the direct Coulomb interaction, and k1 = 1 and k2 = 0for the exchange interaction. The results are as follows:

Wdm SLJG2pe2

h

XK

cKhwJkUKkwJi2F 28

for the dipolemultipole contributions, k2 = K= 2, 4 and 6, where

cK K 1hrKi2

RK2L 2hfkCKkfi21 rK2 29

with c6 ( c4 ( c2, and

Wdd SL1 r12

JG4p

h

e2

R6L

XK

Xe:d:K hwJkUKkwJi2F 30

for the dipoledipole contribution (k2 = 1). If an exchange interac-tion matrix element of the type given by Eq. (16) is used, one gets:

Wex h4fjLi4

J8p3h

e2

R4LhwJkSkwJi2

X

m

/X

j

lzjsmj

/* +

2

F: 31

In these equations SL is the dipole strength of the ligand transitioninvolved in energy transfer process and G is the degeneracy of theligand initial state. In Eq. (31) sm (m = 1, 0, 1) is the spherical com-ponent of the spin operator of electron j in the ligand, lz is the z-component of its dipole operator and S is the total spin operatorof the lanthanide ion. Typical values of the squared matrix elementof the coupled dipole and spin operators in Eq. (31) are in the range10341036 (e.s.u.)2 cm2. This coupled operator breaks down the

usual selection rules in the singletsinglet and singlettriplet ma-trix elements.

Eqs. (30) and (31) differ from the ones obtained in the originaltreatment of intramolecular energy transfer in lanthanide com-

pounds [2]. In the first case, in the previous works the shieldingfactor (1 r1) has been overlooked. In the second case, the ionli-gand overlap has been considered through the shielding factor(1 r0) squared. However, the analysis developed in the abovesections indicates that the correct form should involve h4fjLi4 in

place of (1 r0)2

. This difference introduces a change of approxi-mately three orders of magnitude in the calculated transfer ratesby the exchange mechanism. Nevertheless, estimated ligand-to-ion energy transfer rates by this mechanism are still very fast,assuming values much higher than the typical intraconfigurational4f4f decay rates (radiative and non-radiative). This guarantees thevalidity of the overall conclusions reached in the previous works onthe theoretical emission quantum yields of trivalent europiumcoordination compounds [2].

4. Numerical estimates

4.1. The ionion case

The aim here is to illustrate the critical aspects discussed in theprevious sections, particularly the effect of shielding on the electricmultipolar interaction and the overlap dependence of the exchangeinteraction, and to check their reliability in the evaluation of en-ergy transfer rates. For the ionion case we have chosen a pair oftrivalent ytterbium ions separated by a distance R. Even thoughin this case DJ= 1, as mentioned previously, the magnetic di-poledipole interaction will not be taken into account. Fig. 3 showsthe radial overlap integral between the 4f orbitals of the two ionsas a function ofR. This overlap, S4f, has been calculated from to thefollowing expression:

S24f h4fDj4fAi2

Xn;f

S2nf; 32

where Greek letters now stand for the one-electron quantum num-ber mf in the Slater-type spin-orbitals j4f mf msi. The calculationshave been carried out with the ZINDO program.

The dipoledipole, dipolequadrupole and quadrupolequadru-pole transfer rates were calculated from Eqs. (5), (22), (23) and(24). The oscillator strength of the 2F7/2 ?

2F5/2 transition in theYb3+ ion has the typical value of 3 106. The parity allowed mag-netic dipole contribution to this oscillator strength can be readilyestimated to be 1.2 106. If we assume that half of the remainingoscillator strength value is due to the forced electric dipole mech-

3.0 3.2 3.4 3.6 3.8 4.0

0.0

1.0x10-3

2.0x10-3

3.0x10-3

4.0x10-3

4f-4fradialoverlap

Distance R ()Fig. 3. 4f4f radial overlap integral, as a function of the distance R, calculated fromEq. (32). The individual overlap Sij have been calculated from the ZINDO programwith Slater-type spin-orbitals.



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anism and the other half is due to the dynamic coupling mecha-nism, then we get

PXKhw

JkUKkwJi2 $ 1 1020 cm2 [6]. The ra-dial integral hr2i $ 2 1017 cm2, and the band width at half-height of the 2F7/2 ?

2F5/2 transition is typically of the order of400 cm1. The transfer rate due to the exchange mechanism wasestimated from the crude approximation given in Eq. (15). The re-

sults are collected in Table 1 for R = 3.0, 3.5 and 4.0 .One may note the great difference with respect to the transfer

rates (dd, dq and qq) estimated without the shielding effect(values are given in parentheses). This difference is considerablyaccentuated for the dipoledipole transfer rate. It is also noticeablethe enormous decrease of the exchange transfer rate in going from3 to 4 . For distances higher than 4 this transfer rate is totallynegligible. Another interesting point is that overall the quadru-polequadrupole mechanism dominates. For a distance R = 10 one finds Wdd = 3.2 s

1, Wdq = 17 s1 and Wqq = 1.6 10

2 s1.

4.2. The ionligand case

This case has been intensively studied theoretically in the last

decade, with special attention given to trivalent europium com-plexes. The goal has been to rationalize the mechanisms governingthe intramolecular energy transfer process in these complexes inorder to describe their 4f4f emission quantum yields. Intramolec-ular energy transfer rates both by the direct and exchange mecha-nisms have been calculated for a number of Eu 3+ complexes [2,29].Since our aim here is to emphasize the differences introduced inEqs. (30) and (31) with respect to these previous calculations, wewill focus on order of magnitude values. The necessary quantitiesfor the transfer rate calculations are the same as used previously,except for the introduction of the shielding factor (1 r1)

2 in Eq.(30) and the replacement of (1 r0)

2 by h4fjLi4 in Eq. (31). The ra-dial overlap has been calculated from the following relation [2,19]:

h4fjLi qR0RL

3:5; 33

where q ($0.05) is the same overlap that appears in Eq. (5), and R0is the smallest among the lanthanide ion-ligating atom distances.The selection rules on the total angular momentum quantum num-ber Jof the lanthanide ion are jDJj = 0 or 1, for the exchange mech-anism, and jDJj 6 6 for the dipoledipole and dipolemultipolemechanisms, in both cases J= J0 = 0 excluded. For several com-pounds previously calculated triplet? 5D1 transfer rates by the ex-change mechanism are $1011 1010 s1, while from Eq. (31) thefigures are $108107 s1. For the singlet? 5L6 transfer rates, fromthe dipoledipole mechanism previous values are $109 s1, whilefrom Eq. (30) the figures are $107 s1.

5. Concluding remarks

In lanthanide doped materials (crystals and glasses), when non-radiative energy transfer is operative, several situations that de-

pend on the concentrations of donor and acceptor ions may occur.These situations have been discussed in the literature [9,10]. Ingeneral, the kinetics of the energy transfer process is convenientlytreated by an appropriate system of rate equations and one is ofteninvolved with average energy transfer rates of the following form:

hWETi Z1

R0

WETRGR4pR2 dR; 34

where G(R) is a donor (or acceptor) concentration dependent statis-tical function. Thus, knowing the dependence ofWET on the distanceR, or knowing the relevant energy transfer mechanisms, is a point ofcrucial importance in describing the transfer kinetics. In this re-spect, the theoretical apparatus described in the above sections iscertainly useful.

Not only is the functional dependence ofWET on R of relevancebut also its magnitude, particularly when the different mecha-nisms are competitive. One is then involved with estimates, asaccurate as possible, of donoracceptor distances, 4f radial inte-grals, forced electric dipole intensity parameters, shielding factors

and 4f4f donoracceptor overlap integrals. In this context somepoints have been critically emphasized. The use of experimentalXK intensity parameters in Eqs. (22) and (23) would be an errone-ous procedure, and the shielding effect due to the filled 5s and 5psub-shells cannot be neglected in the direct Coulomb interactionwhen non-antisymmetrized wavefunctions for the donoracceptorpair are used in the evaluation of matrix elements. In the case ofthe exchange interaction, the distance dependence of energy trans-fer rates goes beyond the one usually taken from Dexters formula-tion. Order of magnitude estimates suggest that the exchangemechanism becomes irrelevant for ionion distances larger than4 . We have proposed that more precise estimates of this mecha-nism, including selection rules, can be obtained from the effectiveexchange hamiltonian defined by the matrix element in Eq. (16).

This hamiltonian was used in the case of intramolecular energytransfer (ionligand transfer). Large differences in the calculatedtransfer rates with respect to previous calculations appear. How-ever, since the transfer rates are still much higher than the intra-configurational 4f4f decay rates, these differences will notchange the overall conclusions previously obtained on emissionquantum yields.

Acknowledgements

The author is grateful to the CNPq (Brazilian Agency) and theIMMC and RENAMI (National Projects) for the support. He is alsograteful to Mr Renaldo T. de Moura Jr. for the valuable help withthe calculations of the 4f4f radial overlaps.

References

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Table 1

Estimated Yb3+Yb3+ energy transfer rates

R () Wdd Wdq Wqq Wex (s1)

3.0 4.4 103

(2.8 106)2.6 105

(4.0 107)2.8 107

(1.0 109)2.9 108

3.5 1.7 103

(1.0 106) 7.5 10

4

(1.2 107) 6.0 10

6

(2.3 108) 1.0 10

5

4.0 8.0 102

(5.0 105)2.6 104

(4.0 106)1.6 106

(6.2 107)55

The values in parentheses correspond to the transfer rates without the shieldingfactors.



8/8


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