Hirayama-Inokuti

FIGURE 3. Hirayama-Inokuti plot of relative phosphorescence emission Intensities ( ) and lifetimes (0), under the same conditions as In Fig. 2. Best fit Is obtained for y = 20 (see text) [from figure In J. Polym. Scl., Polym. Chem. Ed. (In press).]... 

TABLE 1. Critical energy transfer distances using Perrin and Hirayama-Inokuti relationships. 

FIGURE 21. Hirayama-Inokuti plot (see text) for triplet-triplet energy transfer from poly(methyl-vinyl ketone) to naphthalene at 77°K as function of naphthalene mole fraction (x) and concentration (c). Solid line theoretical curve for y = 25 [after figure in Eur. Polym. J., 8, 409 (1972)]. 

The shape of the I(t) curves of the donor centers carries very useful information about the nature of the interaction process. Assuming that the acceptors A are randomly distributed at various distances from the donor centers D, the Japanese scientists Inokuti and Hirayama (1965) investigated the shape of the donor decay-time curves for the different multipolar interactions and also for the exchange interaction. 

Inokuti and Hirayama [185] derived these expressions from a rather less formal approach by considering the ensemble-averaged rate of decay of each donor by energy transfer to each acceptor. A more detailed and general analysis has been given by Allinger and Blumen [186], Finally... 

Inokuti-Hirayama Model. The Perrin model is too simplified, although it is convenient for practical use. The static triplet-triplet energy transfer between immobile chromophores dispersed in solids can be well described by the Inokuti-Hirayama theory (17). 

Mataga et al. (18) studied energy transfer from the excited triplet of benzophenone to naphthalene by laser flash photolysis at 77K and showed that the non-exponential decay curves of the benzophenone triplet obey the Inokuti-Hirayama equation (eq (ID). Inokuti and Hirayama (17) themselves compared the data on triplet-triplet transfer between certain aromatic molecules obtained by Terenin and Ermolaev with eq (11) and reported that a good fit was found with an appropriate choice of the parameters C and 7. 

Thus, it is thought that the non-exponential phosphorescence decay of donor in the presence of an acceptor in an Immobilized system is well explained by the Inokuti-Hirayama model based on the static triplet-triplet energy transfer mechanism. This model expects a single-exponential decay for the phosphorescence of a chromophore in the absence of acceptor molecules. However, the phosphorescence decays of organic molecules molecularly dispersed in polymer matrices are known to be non-exponential in some cases even in the absence of other additives. Consequently, other reasons should be considered for such deviations from exponentiallty. 

Due to such uncertainties, and the lack of knowledge about the electronic matrix element, /, many previous analyses of ET phenomena have attempted to ascertain the mechanism from the inter-ion distance dependence of the transfer rate, which is R 10, R 8 and R6 for EQ-EQ, EQ-ED and ED-ED (or MD-MD) ET, respectively. In the Inokuti-Hirayama approach, the donor luminescence intensity as a function of time is given by [361]... 

Inokuti and Hirayama (I-H) (13) developed a quantitative theory of energy transfer by the exchange mechanism and predicted time dependence of fluorescence decay in such coupling. In the 1H approach the S ion is surrounded by a set of A ions at distances RK. 

Inokuti and Hirayama (13) considered the number of activators situated at random in a sphere around a sensitizer such that the activator concentration is constant when the volume of the sphere and the number of activator ions considered goes to infinity. They obtained the following expression for the intensity decay of the emission of the sensitizer surrounded by many activators ... 

Watts and Richter (36) showed that the luminescence decay of the level of Yb3+ in YF3 with 0.3% Yb3+, 6% Ho3+ fits the theoretical curve for dipole-dipole interaction. Reisfeld and Boehm (28) have fitted the decay curve of Sm3+ in phosphate glasses to the theoretical curve of Inokuti and Hirayama and found that the quenching interaction arises from the d—q mechanism. 

---