Asymptotic behavior of fluorescence line narrowing in dilute systems with finite-range transfer

Journal of Luminescence 31 & 32 (1984) 645-647 North-Holland, Amsterdam

645

ASYMPTOTIC BEHAVIOR OF FLUORESCENCE LINE NARROWING IN DILUTE SYSTEMS WITH FINITE—RANGE TRANSFER

0. L. HUBER

Department of Physics, University of Wisconsin, Madison, WI 53706

We diacuaa the long—time behavior of the decay of the sharp line component observed in studies of dilute systems with finite—range energy transfer carried out with fluorescence line narrowing techniques. It is shown that the asymptotic intensity is a geometrical quantity related to the distribution of finite clusters. Detailed results are presented for the square and simple cubic lattices with nearest—neighbor transfer along with a three dimensional continuum model.

The purpose of this paper is to analyze the long—time behavior of 1’2 in dilute systems with finite—range energy fluorescence line narrowing transfer. The work discussed is an extension of an earlier study of fluorescence line narrowing in systems with nearest—neighbor transfer.3 The main point we wish to make is that in systems where the transfer rate between ions separated by more than a fixed distance is zero, the asymptotic behavior of the intensity of the narrowed component of the fluorescence following excitation of an inhomogeneously broadened line by a narrow—bançl source is determined by the distribution of finite clusters.

[A cluster is defined as a

set of optically active ions connected by non—zero transfer rates; transfer can not take place between ions belonging to different clusters.] Since the cluster distribution is a purely geometrical property, one can determine the asymptotic behavior without reference to the microscopic details of the energy transfer process. The dependence on the properties of the finite clusters follows from the observations that if, initially, a cluster of a ions has a single ion in the excited state the asymptotic probability of the same ion being excited is s (assuming symmetric transfer rates, as is appropriate when kT>> mismatch in energy).

In contrast, an initially excited ion belonging to an infinite

cluster has a vanishingly small probability of being excited in the long—time limit. The analysis of Ref.

3 pertained to a situation where the ions were located

on a lattice with transfer limited to ions occupying nearest—neighbor sites.

0022—2313/84/$03.OO© Elsevier Science Publishers By. (North-Holland Physics Publishing Division)

D. L. 1-tuber / Asi’rnptotic behavior of fluorescence line narrowing

646

We write 1(t)

the intensity of the narrowed component of the fluorescence, as

,

1(t) where

e

—1 t ~ 1(0)1(t),

is the radiative rate.

(1) The asymptotic limit of 1(t), I(~),can be

expressed as a sum over clusters of the probability of an ion belonging to a cluster of s ions multiplied by s~, Ref.

For the lattice models discussed in

3 I(~) takes the form I(~)

In (p),

(2)

sri ~ where p is the fraction of lattice sites occupied by optically active ions and n

5(p)

is the number of clusters of size s per occupied site.

Using cluster data provided by ~

we have calculated I(r~) as a

function of p for square and simple cubic lattices.

~

~O~2’O~4’+

O~8

FIGURE 1 i(~) vs p for the square lattice with nearest—neighbor transfer. The arrow denotes the critical percolation concentration,

Figs.

O.2~

in

O~6

FIGURE 2 i(~) vs p for the simple cubic lattice with nearest—neighbor transfer. The arrow denotes the critical percolation concentration.

1 and 2. From the figures it is evident that I(~) falls smoothly with

increasing p, being equal to 1 at p at p

The results are shown

1 (no finite clusters).

0 (each ion in a cluster of size 1) and 0

At the critical percolation concentration,

which marks the first appearance of the infinite cluster, I(~) is equal to 0.05 for the square lattice and 0.17 for the simple cubic structure. One can also analyze the situation where the maximum range of transfer is large in comparison with the minimum spacing between ions.

Under these

conditions a continuum model is appropriate where the concentration is expressed in terms of the dimensionless variable ~ defined by P

~1pr~.

(3)

DL. Huher / Asymptotic behavior of fluorescence line narrowing

647

In this equation p is the number of optically active ions per unit volume and rM is the maximum distance over which transfer can take place. The asymptotic value of 1(t) can be written I(~) = ( I sC (pfl~, sri

(‘U

~

where C

5(p) is the fraction of clusters of s ions. We have calculated 1(w) 5 The over the range O
a2~~

FIGURE 3 vs ~ for a three dimensional continuum. Here p = (4vr/3)pr3, where p is the concentration of optically active ions and rM is the max~mumrouge for energy transfer. ~c is approximately equal to 2.7. I(oo)

ACKNOWLEDGMENT We would like to thank U. F. Edgal and 2. D. Wiley for providing us with the original data displayed grsphically in their paper.

The research was supported

by the National Science Foundation under the grant DMR—8203704. REFERENCES 1) P. M. Selzer, Laser Spectroscopy of Solids Topics in Applied Physics, edited by W. N. Yen and P. N. Selzer (Springer, Berlin,

1981), Vol. ~49,Chap. 4.

2) 0. L. Huber, 0. 5. Hamilton, and B. Bsrnett, Phys. Rev. 816, 4642 (1977).

3) 0. L. Huber, 2. Chem. Phys. 81 (1984). 4) A. Flsmmang, Z. Phys. 828, 47 (1977). 5) U. F. Edgal and 2. 0. Wiley, Phys.

Rev. 827, 4997 (1983).