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Relationship Between Absorption Intensity and Fluorescence Lifetime of Molecules S Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules S. J. Strickler and Robert A. Berg Citation: J. Chem. Phys. 37, 814 (1962); doi: 10.1063/1.1733166 View online: http://dx.doi.org/10.1063/1.1733166 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v37/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 13 May 2012 to 139.18.118.45. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions THE JOURNAL OF CHEMICAL PHYSICS VOLUME 3;, ~UMBER 4 AUGUST 15,1962 Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules* S. ]. STRICKLER AND ROBERT A. BERG Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley, California (Received March 12, 1962) The equations usually given relating fluorescence lifelime to absorption intensity are strictly applicable only to atomic systems, whose transitions are sharp lines. This p3.per gives the derivation of a modified formula l/ro= 2.880X 10-9 n'(vr)A,-l(!(I/g,,) fEd lnv, which should be valid for broad molecular bands when the transition is strongly allowed. Lifetimes calculated by this for;nula have been comp3.rd with measure:llifeti~e5 for a number of organic molecules in solution. In most cases the values agree within experi nental error, indicating that the formula is valid for such systems. The limitation3 of the for.nula an:l the r ~s:.llts expecteJ for weak or forbidden transitions are also discussed. N 1917, Einstein! derived the fundamental relation­ absorption. This means that, in general, the equation I ship between the transition probabilities for induced is strictly applicable only to atomic transitions. There absorption and emission and that for spontaneous are many problems of interest, however, in which it is emission. The result showed that the spontaneous­ desirable to calculate the mean life of excited states of emission probability was directly proportional to the molecules, for which Eq. (1) is not accurate. Lewis corresponding absorption probability and to the third and Kasha4b tried to test the equation approximately power of the frequency of the transition. Einstein's for rhodamine B, and concluded that it was probably equations were soon expressed in terms of more com­ valid to within a factor of two. Bowen and Metcalf,5 in monly measured quantities by Ladenburg2 and Tolman.3 discussing the fluorescence of anthracene, gave an Later a refractive index correction was added by equation similar to (1) except for an extra factor of 3 Perrin4a and by Lewis and Kasha4b for cases where the on the right-hand side. They suggested that this factor absorbing systems were in solution. Although the was necessary for molecules which absorb light only various authors expressed the result in different ways, when a component of the electric vector of the light all their equations may be written in the form wave is oriented along one particular axis of the molecule. Our work will show that this factor is 1/ro= A,,_l= 8X23031rcVUI2n291-I.f:jfdv. (1) erroneous. Other authors6 have pointed out that it is gu necessary to take account of the frequency difference between absorption and fluorescence. However, they In this equation, A u_ 1 is the Einstein transition probability coefficient, or rate constant, for spontaneous have not given the result in exact form. emission from an upper state u to the lower state I It is the purpose of this paper to present a modifica­ (usually the ground state); C is the speed of light in a tion of Eq. (1) which is applicable to polyatomic I molecules under certain conditions. We shall first vacuum; Vul is the frequency of the transition in cm- ; n is the refractive index of the medium; 91 is Avogadro's derive the equation, and in doing so, for the sake of number; gl and gu are the degeneracies of the lower completeness, we shall repeat the essential points of the and upper states, respectively; f is the molar extinction derivation of the earlier formulas. We shall then coefficient. The integration extends over the absorption summarize calculations of the fluorescent lifetimes of a band in question. TO, the reciprocal of the rate constant, number of molecules, and describe the experiments by is the maximum possible mean life of state u, i.e., the which we measured these lifetimes. Finally, we shall mean life if the only mechanism of deactivation is discuss the limitations of the formula and the cases spontaneous emission to state I. The actual mean life T where it may not be valid. may be shorter than TO if the quantum yield of the DERIVATION OF EQUATION FOR MOLECULES luminescence is less than one. In the derivation of Eq. (1), it is necessary to Consider two electronic states of a molecule, a assume that the absorption band is sharp, and that the ground state I and an upper state u. The corresponding fluorescence occurs at the same wavelength as the electronic wave functions may be called 8 1 and 8 u . * This work was carried out under the auspices of the U. S. 5 E. ]. Bowen and W. S. Metcalf, Proc. Roy. Soc. (London) Atomic Energy Commission. A206,437 (1951). 1 A. Einstein, Physik. Z. 18, 121 (1917). 6 T. Forster, Fluoreszenz Organischer Verbindungen (Vanden- 'R. Ladenburg, Z. Physik 4,451 (1921). hoeck & Ruprecht, Gottingen, 1951), p. 158; F. E. Stafford, 3 R. C. Tolman, Phys. Rev. 23,693 (1924). Ph.D. dissertation, University of California, Berkeley, 1959, 4 (a) F. Perrin, J. phys. radium 7,390 (1926); (b) G. N. Lewis UCRL-8854; V. N. Soshnikov, Soviet Phys.-Uspekhi 4, 425 and M. Kasha, J. Am. Chern. Soc. 67, 994 (1945). (1961). 814 Downloaded 13 May 2012 to 139.18.118.45. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions FLUORESCENCE LIFETIME OF MOLECULES 815 Associated with each electronic level are a series of convenience that the light beam has a cross section 1 cm2 states having different wave functions for nuclear with a uniform intensity over this area. If p(v, x) is the motion, vibration, rotation, and translation. For the radiation density in the light beam after it has passed present purpose it is sufficient to consider only the a distance x centimeters through the sample, the molar vibrational states, though the rotational and transla­ extinction coefficient E can be defined by tional motion might be treated in an analogous manner. p(v, x) / p(v, 0) = 10-0(.)0",= e-2.3030(v)0"" In the Born-oppenheimer approximation, each vibronic state has a wave function'!t which can be written as a where C is the concentration in moles per liter. product of an electronic function 8 and a vibrational If a short distance dx is considered, the change in function <1>. '!t za =8z<I>la, '!tub=8,,<I>ub. For simplicity in radiation density may be written discussion, it will be assumed that each state is single; if degeneracies occur in either the electronic or vibra­ -dp(v) =2.303E(V)p(V, 0) Cdx. (8) tional part, they can be summed over in the proper For simplicity, all the molecules will be assumed to be manner when necessary. in the ground vibronic state, '!t zo • (This may not be true at normal temperatures, but this will not materially Relationship Between Einstein A and B Coefficients affect the results. It would be possible, but more Suppose a large number of these molecules, immersed complicated, to carry through a number of states with in a nonabsorbing medium of refractive index n, to be appropriate Boltzmann weighting factors.) It is easily in thermal equilibrium within a cavity in some material seen that 1 at temperature T. The radiation density (erg cm-3 per Cdx= 1000NlOin- • (9) unit frequency range) within the medium is given by Furthermore !::.N (v), the number of molecules excited Planck's blackbody radiation law per second with energy hv, is given by p(v) = (87rhv3n3jc3) [exp (hv/k T) _1]--1. (2) !::.N(v)=-cdp(v)/(hvn). (10) By the definition of the Einstein transition proba­ Combining Eqs. (8), (9), and (10), it is found that bility coefficients, the rate of molecules going from state !::.N(v)/NIO=[2303ce(v)/hvnin]p(v, 0). (11) la to state ub by absorption of radiation is Equation (11) gives the probability that a molecule (3) in state to will absorb a quantum of energy hv and go to some excited state. To obtain the probability of where N la is the number of molecules in state la, and going to the state ub, it must be realized that this can V~ub is the frequency of the transition. Molecules in occur with a finite range of frequencies, and Eq. (11) state ub can go to state la by spontaneous emission must be integrated over this range. p(v, 0) can be with probability Aub-+Za or by induced emission with assumed constant over this range and equal to p(VID-ub). probability Bub-+ZaP(Vub-+la)' The rate at which molecules The value of E(V) must be only that for the one vibronic undergo this downward transition is given by transition; if the spectrum is well resolved, this would (4) present little difficulty, but it can be done in principle in any case.
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