Applicability of the shell model for energy transfer

Journal of Alloys and Compounds 275–277 (1998) 250–253

L

Applicability of the shell model for energy transfer a, a b Thomas Luxbacher *, Harald P. Fritzer , Colin D. Flint a

b

¨ Physikalische und Theoretische Chemie, Technische Universitat ¨ Graz, Rechbauerstraße 12, 8010 Graz, Austria Institut f ur Laser Laboratory, Department of Chemistry, Birkbeck College, University of London, 29, Gordon Square, London WC1 H 0 PP, UK

Abstract In the cubic hexachloroelpasolite crystals Cs 2 NaSmx Eu y Gd 12x 2y Cl 6 the emission from the 4 G 5 / 2 state of Sm 31 is strongly quenched by cross-relaxation and energy-transfer processes. At temperatures above 100 K the energy-transfer rate consists of non-resonant contributions due to electric dipole vibronic–electric dipole vibronic interaction and a near-resonant contribution where both the donor and acceptor transitions are magnetic-dipole allowed. The near-resonant energy-transfer rate is calculated and the small discrepancy between the theoretical and experimental values shows the applicability of the shell model for highly symmetric crystals.  1998 Elsevier Science S.A. Keywords: Energy-transfer rate; Hexachloroelpasolite crystals; Shell model

1. Introduction In a previous paper we applied a shell model for the simultaneous treatment of cross-relaxation and energytransfer processes to understand the decay kinetics of the 4 G 5 / 2 state of Sm 31 in the cubic hexachloroelpasolite crystals Cs 2 NaSm x Eu y Gd 12x 2y Cl 6 [1]. In this system the emission from the 4 G 5 / 2 state of Sm 31 is strongly quenched by both cross-relaxation to nearby Sm 31 ions and energy transfer to the 5 D 0 state of Eu 31 acceptors. The temperature dependence of the intrinsic decay rate k 0 of an isolated Sm 31 ion and the cross-relaxation rate k CR to a single nearest-neighbour Sm 31 acceptor can be described by the coth law for vibronic transitions whereas the energy-transfer rate k ET from the Sm 31 donor ion to a single Eu 31 acceptor in the first shell shows a rapid increase at temperatures above 100 K [2]. The stronger temperature dependence of this rate indicates that two different energy-transfer processes are present. At low temperatures the increase of k ET is consistent with a squared coth law for electric dipole vibronic–electric dipole vibronic interaction whereas near-resonant transfer involving the Sm 31 ( 4 G 5 / 2 )G7 →( 6 H 7 / 2 )G8 donor transition and the Eu 31 ( 7 F 1 )G4 →( 5 D 0 )G1 acceptor transition shown in Fig. 1 as a pair of transitions BB9 becomes dominant as

the temperature is raised where both transitions are of magnetic-dipole allowed electronic origin. The total energy-transfer rate to a single Eu 31 acceptor in the first shell shows an almost linear increase in the temperature range 200 to 300 K which is well understood in terms of the thermal population of the initial Sm 31 ( 4 G 5 / 2 )G7 donor and Eu 31 ( 7 F 1 )G4 acceptor states [2]. The energy gaps between the G8 and G7 levels of the 4 G 5 / 2 state of Sm 31 as well as between the 7 F 0 and 7 F 1 states of Eu 31 are about 350 cm 21 , such that the near-resonant process involving these initial states contributes to the total energy-transfer rate at temperatures .100 K. In this paper we calculate the near-resonant contribution to the energy-transfer rate between a Sm 31 donor ion and a ˚ assuming magnetic Eu 31 acceptor at a distance R 1 57.65 A dipole–magnetic dipole interaction among donor and acceptor ions. Within the theoretical uncertainty and the experimental error the order of magnitude of the calculated energy-transfer rate is in agreement with the near-resonant contribution to the experimental rate which is related to the exponential decay rate of luminescence from the 4 G 5 / 2 state of Sm 31 in Cs 2 NaSm 0.01 Eu 0.99 Cl 6 [1].

2. Theory

2.1. The shell model for energy transfer *Corresponding author. [email protected]

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0925-8388 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 98 )00314-4

Within our shell model assuming electric dipole (vib-

T. Luxbacher et al. / Journal of Alloys and Compounds 275 – 277 (1998) 250 – 253

251

Fig. 1. Energy levels and transitions involved in the cross-relaxation and energy-transfer processes in Cs 2 NaSm x Eu y Gd 12x 2y Cl 6 . The letters indicate pairs of near-resonant transitions.

ronic)–electric dipole (vibronic) and / or magnetic dipole– magnetic dipole interaction among donor and acceptor ions the donor emission decay following a d-function excitation pulse takes the form [1] shells Nn Nn 2r n

I(t) 5 I(0)exp(2k 0 t) exp

F

PO OO

n51 r 50 q 50 n n

Nn r n, qn

(x,y)

S DG

R1 2 CR ET 2 ](r n k 1 qn k ) ] 3 Rn

6

t

(1)

In the present system, k 0 is the intrinsic decay rate of an isolated Sm 31 ion including radiative and non-radiative relaxation processes, k CR is the cross-relaxation rate from the donor ion to a chemically identical acceptor in the first ET shell, and k is the rate of energy transfer to a n51 acceptor chemically different from the donor. The occupancy factor O Nr nn, qn (x,y) is the probability of finding r n acceptor ions chemically identical to the donor and qn chemically different acceptors in the nth shell at a distance R n which has a capacity to contain Nn acceptors [1]. Since the rates k 0 , k CR , and k ET are determined from exponential decay of the Sm 31 ( 4G 5 / 2 ) emission in Cs 2 NaSm 0.001 Gd 0.999 Cl 6 , Cs 2 NaSmCl 6 , and Cs 2 NaSm 0.01 Eu 0.99 Cl 6 , respectively, the donor decay curves for all other concentrations of donor and acceptor ions may be calculated by Eq. (1) without any adjustable parameters.

2.2. Resonant energy transfer For energy transfer between rare-earth ions we consider transitions from the initially excited crystal-field component u[a 9D J 9D ]G 9Dg 9D l of multiplet u[a 9D J 9D ]G 9D l to the lower lying crystal-field component k[aD JD ]GDgD u of multiplet k[aD JD ]GD u of the donor ion with a similar notation for the acceptor. a represents any other quantum number necessary to specify the state. The total crystal-field splitting is typically a few cm 21 and the population of the component levels is given by a Boltzmann distribution as pd 9 5 gd 9 exp(2Ed 9 /k B T )

FO g

d0

exp(2Ed 0 /k B T )

d0

G

21

(2)

where the energy Ed 9 is related to the position of the lowest level of the multiplet taken as Ed 0 5 0, and gd 9 is the degeneracy of the d9th level. The resonant energy-transfer rate from a donor ion to an acceptor then becomes [3,4] k

ET

O

E

2p 2 5 ] pd 9 pa ukda9uHˆ DA ud9alu fd 9d (E)faa9 (E)dE " d 9,a

(3)

For magnetic dipole–magnetic dipole interaction the Hamiltonian in Eq. (3) takes the form

m m 3( m R)( m R) O F]]] 2 ]]]]] G R R

m0 MD Hˆ DA 5 ] 4p i, j

i d 9d

j aa9

3

i d 9d

j aa 9

5

(4)

where m id 9d is the ith component of the magnetic-dipole

T. Luxbacher et al. / Journal of Alloys and Compounds 275 – 277 (1998) 250 – 253

252

moment of the donor transition and i and j running over x, y, and z. Rewriting Eq. (4) the transition matrix element is given by m0 2 MD 2 i 2 2 j 2 ukda9uHˆ DA ud9alu 5 ]]3 ( m d 9d ) (Cij ) ( m aa 9 ) (5) 4p R i, j

S

DO

In a face-centred cubic lattice, the components of the magnetic-dipole moment are independent of orientation i 2 2 such that ( m d 9d ) 5 ]31 ( md 9d ) , etc. Introducing a geometric factor which includes the angular dependence of the interacting magnetic dipoles as 1 G MD d 9d;aa9 5 ] 9

O(C )

2

(6)

ij

i, j

Eq. (5) becomes

m0 ukda9uHˆ DA ud9alu 5 ]]3 4p R MD

S

2

DG 2

MD d 9d;aa9

2

( md 9d ) ( maa 9 )

2

(7)

Writing the crystal-field states as

Oc(JMuJGg )u f

u f N [a SLJ]Gg l 5

N

[a SL]JMl

(8)

M

etc., the g0 th component of the magnetic-dipole transition moment transforming as the representation G4 of the octahedral double group 2 O is given by 4g0 m Gd 9d 5

Okf

g ,g 9

N

4) ˆ (g1G [a SLJ]Gg um u f N [a 9S9L9J9]G 9g 9l 0

(9)

ˆ (1G4 ) 5 mB (Lˆ 1 2Sˆ ) is the magnetic moment tensor where m operator and mB is the Bohr magneton. Application of the Wigner–Eckart theorem to the matrix element in Eq. (9) gives [5] 4) ˆ (g1G k f N [a SLJ]Gg um u f N [a 9S9L9J9]G 9g 9l 0

S

5

JG g

1G4 g0

D

J9G 9 ˆ ( 1 ) i f N [a 9S9L9]J9l k f N [a SL]Jim g9 (10)

within the hJ,G j scheme. The symmetry coupling coefficients

S

JG g

1G4 g0

D O

S

J9G 9 J9 1 5 (21) J 92M 9 g9 2 M9 q M,M 9, q

c(1qu1G4g0 )c(JMuJGg )c(J9M9uJ9G 9g 9)*

D

J M

Cs 2 NaSm 0.01 Eu 0.99 Cl 6 show a fast non-exponential initial decay due to cross-relaxation to nearest-neighbour Sm 31 acceptor ions with a rate of k CR 52120 s 21 . At times .2.5310 23 s both decay curves are essentially exponen4 31 tial with the lifetime of the G 5 / 2 state of Sm decreasing 31 31 by a factor of 6.6 for replacing Gd by Eu . Both decay curves are exactly described by Eq. (1) and the stronger quenching of luminescence from the 4 G 5 / 2 state in Cs 2 NaSm 0.01 Eu 0.99 Cl 6 is assigned to additional energy transfer between Sm 31 donor ions and Eu 31 acceptors at a rate k ET 549.9 s 21 .

3.2. Numerical calculation In the following we calculate the near-resonant magnetic dipole–magnetic dipole contribution to the energy-transfer rate from a Sm 31 donor ion to a single Eu 31 acceptor ion in the first shell. The calculation of resonant energy¨ transfer rates using the Forster–Dexter equation for magnetic dipole–magnetic dipole interaction [3,4] involves the determination of crystal-field eigenstates for both donor and acceptor ions and the magnetic dipolar coupling strength associated with the transitions considered in the energy-transfer process. The energy mismatch between donor emission and acceptor absorption transitions is introduced in the overlap integral of normalized lineshape functions for these transitions. 31 The compositions of the free-ion energy levels of Sm 31 4 6 and Eu involved in the ( G 5 / 2 )G7 →( H 7 / 2 )G8 donor 7 5 transition and the ( F 1 )G4 →( D 0 )G1 acceptor transition shown in Fig. 1 as a pair of transitions BB9 are taken from Refs. [7–9], respectively. The basis functions used in the calculation of magnetic-dipole transition moments are taken from Griffith [10]. We note that beside these resonant transitions a non-resonant pathway involving the ground states of both the initial donor and acceptor multiplets is present and indicated as a pair of transitions AA9 in Fig. 1. The magnetic dipolar coupling strength for the donor transition is given by ˆ (1G4 ) u( 4 G 5 / 2 )G7 l 5 k( 6 H 7 / 2 )G8 um

OO

S,S 9J, J 9 L,L 9

(11)

are readily calculated for the various JG → J9G 9 electronic transitions involved in the energy-transfer process considered in this paper. The reduced matrix elements in Eq. (10) are calculated using Eqs. (10)–(13) of Ref. [6].

3. Results

3.1. Experimental decay curves At 300 K the luminescence decay curves from the 4 G 5 / 2 state of Sm 31 in both Cs 2 NaSm 0.01 Gd 0.99 Cl 6 and

ˆ k[SL]Jim

(1)

i[S9L9]J9ldSS 9dLL 9 D(JJ91)

O

g ,g 9,g0

S

c(JG8g )c(1G4g0 )c(J9G7g 9)

D

JG8 1G4 J9G7 g g0 g9

(12)

The magnetic-dipole moments for the ( 4 G 5 / 2 )G7 →( 6 H 7 / 2 )G8 donor transition and the ( 7 F 1 )G4 →( 5 D 0 )G1 acceptor transition are (1.83060.497) m B and 2(0.46360.126) m B , respectively. Substitution of these values in Eq. (7) and utilizing the geometric factor for magnetic dipole–magnetic dipole interaction, G 6n 5 ]23 [11], the square of the interaction energy representing the coupling of the magnetic dipoles

T. Luxbacher et al. / Journal of Alloys and Compounds 275 – 277 (1998) 250 – 253

md 9d and maa9 is E 2int 5 (1.96361.066) 3 10 253 J 2 . The resonant energy-transfer rate k ET between the Sm 31 donor ion and a nearest-neighbour Eu 31 acceptor is related to this interaction energy by

E

2p k ET (MD) 5 ] p( 4 G 5 / 2 )G7 p( 7 F 1 )G4 E 2int fd 9d (E)faa 9 (E)dE " (13) where the overlap integral is calculated using normalized Lorentzian line-shape functions for the donor emission and acceptor absorption transitions. The full width at half maximum, DE, is taken as 9.932310 223 J for both donor and acceptor transitions, and the transition maxima 219 are measured as (3.35260.001)310 J and 219 (3.34960.002)310 J for donor and acceptor transitions, respectively, corresponding to (1687268) and (1685865) cm 21 . Considering these values and the integral of product Lorentzian functions which is evaluated as

Ef

DEd 9d 1 DEaa9 2 ]]]]]]]]]] d 9d (E)faa9 (E)dE 5 ] p 4(Ed 9d 2 Eaa9 )2 1 (DEd 9d 1 DEaa 9 )2 (14)

the value of the overlap integral is 3.625310 220 J 21 . At 300 K the thermal population of the Sm 31 ( 4 G 5 / 2 )G7 state and the Eu 31 ( 7 F 1 )G4 level are about 8 and 32%, respectively. Substitution in Eq. (13) gives k ET 5(12.066.5) s 21 for the rate of near-resonant energy transfer between a Sm 31 donor ion and a single nearest-neighbour Eu 31 acceptor ion at 300 K.

4. Conclusions We have calculated the near-resonant contribution, k ET 5 (12.066.5) s 21 , to the rate of energy transfer from a Sm 31 donor ion to a nearest-neighbour Eu 31 acceptor in the hexachloroelpasolite crystals Cs 2 NaSm x Eu y Gd 12x 2y Cl 6 assuming magnetic dipole–magnetic dipole interaction among donor and acceptor ions. Comparison of this rate

253

with the total energy-transfer rate, k ET 549.9 s 21 , determined experimentally from the luminescence decay curve of the 4 G 5 / 2 state of Sm 31 in Cs 2 NaSm 0.01 Eu 0.99 Cl 6 by using the shell model for energy transfer shows a discrepancy by a factor of 4. Beside the near-resonant magnetic dipole–magnetic dipole contribution the experimental rate consists of additional non-resonant electric dipole vibronic–electric dipole vibronic contributions which are not included in our calculation. Within the uncertainty involved in the theoretical calculation the order of magnitude of the near-resonant contribution to the total energy-transfer rate is comparable with the experimental value estimated from its temperature dependence as (3065) s 21 . This agreement shows the applicability of the shell model to determine energy-transfer rates in crystalline solids of high symmetry.

Acknowledgements One of us (TL) acknowledges partial financial support of this work by the Graz Technical University.

References [1] [2] [3] [4] [5] [6] [7] [8]

[9] [10] [11]

T. Luxbacher, H.P. Fritzer, C.D. Flint, J. Lumin. 71 (1997) 177. T. Luxbacher, H.P. Fritzer, C.D. Flint, J. Lumin. 71 (1997) 313. ¨ T. Forster, Ann. Phys. 2 (1948) 55. D.L. Dexter, J. Chem. Phys. 21 (1953) 836. ¨ E. Konig, S. Kremer, Int. J. Quant. Chem. 8 (1974) 347. W.T. Carnall, P.R. Fields, B.G. Wybourne, J. Chem. Phys. 42 (1965) 3797. B.G. Wybourne, J. Chem. Phys. 36 (1962) 2301. The parameters to calculate the composition of the 4 G 5 / 2 free-ion state of Sm 31 are taken from D.R. Foster, F.S. Richardson, R.W. Schwartz, J. Chem. Phys. 82 (1985) 618. G.S. Ofelt, J. Chem. Phys. 38 (1963) 2171. J.S. Griffith, The Theory of Transition Metal Ions, Cambridge University Press, Cambridge, MA, 1964. S.O. Vasquez, C.D. Flint, Chem. Phys. Lett. 238 (1995) 378.