Solvent Effects on C 60 Excited State Cross Sections
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Mat. Res. Soc. Symp. Proc. Vol. 374 01995 Materials Research Society
Table I. The solution properties and nonlinear optical properties. A" To CG Solvent Concentration Solubility Limit mg/mi (Tol Ref) xl0 1 9 cm 2 x10 7cm 2 xl0 m (mM) Abbreviation 1.32 1.32 0.949 0.40 2.30 1.66 BZ 0.65 0.65 0.950 0.48 1.59 TOL 1.78 2.9 0.40 2.96 0.930 1.30 2.17 EBZ 2.13 0.35 2.13 0.893 5.41 3.90 PXY 2.41 0.37 2.4 0.891 3.52 4.89 MXY 2.10 0.42 2.1 4.75 0.844 6.59 OXY 2.05 0.17 2.05 2.50 0.966 3.37 CB 0.52 0.83 0.52 1
19.5
1.11
0.959
0.61
differences between the integrating sphere measurement and the standard absorbance measurement indicate that is necessary to use integrating sphere measurements to correctly determine the internal absorption and ameliorate the effects of scattering. The resulting ground state absorption cross sections,UG, at 694 nm are also reported in Table I. Errors in the measurement of the ground state cross section arise from either the concentration determination or the absorbance Transmittance (transmittance) measurement. measurements are accurate to • 1%. However the effect of this error is magnified at high transmittances. The concentration measurements are accurate to •10%. Overall the ground state cross sections are accurate to
r om
330 135 742 602 651 498 1214 63 -
11 36
Figure 1. The energy levels and transitions modeled.
±_15%.
MODEL A rate equation model for C60 is developed which includes the levels and processes shown in Figure 1. Relaxation from the highest excited states is assumed to be much faster than the excitation rate, and thus the populations in these states is negligible. Consequently the populations in three levels and the radiation transport are modeled, dG dt
dS dt
aGIG
ho
UGIG -S=CGI
-
S
(kS + ki-sc ) S
ho) dT=k. S-kTT
=
U9GGIN -(asSS+
294
(2)
(3)
dt dI
(I)
TT)IN
(4)
where G, S, and T represent the fraction of the total number density, N, in the ground state, first excited singlet state, and the lowest triplet state, respectively. The rates, k, and cross sections, U, are associated with the transitions shown in Figure 1. These are solved in the nondepeletion limit such that G = 1, for a square temporal pulse of peak irradiance I and pulse width T,and assuming rol/kT. With these approximations all of the temporal equations can be solved and the radiation transport equation integrated over time to give, --
dz
U=_CNF -(U 2 G
eff
-
`2(5
(5)
UG ) NF'
where, F' = F/FG, FG = hw0/ UG, and, Acr=C~~eff-•rG =7 2UUf-U
'G]r-2
-{-Y-(TCT sinh •e-
/2
+ OT(
T-CrG) ±•-}
(6)
and r7= T(ks+kisc),'T =kisc/(kisc+ks). For the short pulse case ofc(kS + kis)-r)the Ueff= CFTO. Integration of equation (5) assuming a Gaussian spatial distribution and collimated radiation, results in the nonlinear energy transmittance, In (I+F /F)(7 T =To
FP/FC
(7)
where Fp is the peak on-axis fluence, To is the linear internal absorption driven transmission and,
2ho)
aU(l - T0)
(8)
These expressions reveal that the critical fluence, Fc, where the transmittance becomes nonlinear,
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