As the assessment for the optical spectroscopy module, write a report (10 A4 page limit, font Arial 12, line spacing 1.5) to investigate the effectiveness of the Strickler &Berg formula

(J. Chem. Phys. 37, 814-822, 1962) to estimate the natural lifetime of Anthracene in solution. This work will involve several different tasks, and is the only work you have to do in this module. You will be assessed in the different tasks involved in the application of the Strickler & Berg formula to obtain an estimative of the fluorescence lifetime, as well as on the quality of the manuscript. Your report must include the typical sections in a research paper: abstract, introduction, methods, results and discussion, conclusions and references. An excel file with UV-Visible absorption and emission data of Anthracene is included. These data was obtained in the following way: A toluene stock solution was prepared by dissolving 1.36 mg of Anthracene (Mr = 178.23 gmol-¹) in 6 ml of toluene. Four solutions were subsequently prepared by diluting the stock solution by a Vi factor (V/v₁) equal to: solution 1= 0.01, solution 2=0.02, solution 3=0.06 and solution 4-0.09, where V; represents the initial volume of the stock solution, and Vf is the volume of the diluted solution. ∞ Fig. 1 Anthracene molecule./n The Journal of Chemical Physics AIP Publishing RESEARCH ARTICLE | JULY 20 2004 Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules S. J. Strickler; Robert A. Berg Check for updates J. Chem. Phys. 37, 814–822 (1962) https://doi.org/10.1063/1.1733166 View Export Online Citation APL Machine Learning Latest Articles Online! Read Now CrossMark AIP Publishing 04 January 2024 10:42:39 THE JOURNAL OF CHEMICAL PHYSICS Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules* S. J. 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 lifetime to absorption intensity are strictly applicable only to atomic systems, whose transitions are sharp lines. This paper gives the derivation of a modified formula 1/ro=2.880×10-³ n²(ỹƒ¯³) av¯¹ (81/gu) fed Inỹ, which should be valid for broad molecular bands when the transition is strongly allowed. Lifetimes calculated by this formula have been compared with measured lifetimes for a number of organic molecules in solution. In most cases the values agree within experimental error, indicating that the formula is valid for such systems. The limitations of the formula and the results expected for weak or forbidden transitions are also discussed. absorption and emission and that for spontaneous emission. The result showed that the spontaneous- emission probability was directly proportional to the corresponding absorption probability and to the third power of the frequency of the transition. Einstein's equations were soon expressed in terms of more com- monly measured quantities by Ladenburg² and Tolman.³ Later a refractive index correction was added by Perrina and by Lewis and Kashab for cases where the absorbing systems were in solution. Although the various authors expressed the result in different ways, all their equations may be written in the form I ship between the transition probabilities for induced 1917, Einstein¹ derived the fundamental relation- absorption. This means that, in general, the equation is strictly applicable only to atomic transitions. There are many problems of interest, however, in which it is desirable to calculate the mean life of excited states of molecules, for which Eq. (1) is not accurate. Lewis and Kashab tried to test the equation approximately for rhodamine B, and concluded that it was probably valid to within a factor of two. Bowen and Metcalf,5 in discussing the fluorescence of anthracene, gave an equation similar to (1) except for an extra factor of 3 on the right-hand side. They suggested that this factor was necessary for molecules which absorb light only when a component of the electric vector of the light wave is oriented along one particular axis of the molecule. Our work will show that this factor is erroneous. Other authors have pointed out that it is necessary to take account of the frequency difference between absorption and fluorescence. However, they have not given the result in exact form. 1/70= Au»l=8×2303πcũul³n²N-1³1 [ +-181 fedő. gu (1) In this equation, Au» is the Einstein transition probability coefficient, or rate constant, for spontaneous emission from an upper state u to the lower state l (usually the ground state); c is the speed of light in a vacuum; v is the frequency of the transition in cm-¹; n is the refractive index of the medium; N is Avogadro's number; g, and gu are the degeneracies of the lower and upper states, respectively; e is the molar extinction coefficient. The integration extends over the absorption band in question. To, the reciprocal of the rate constant, is the maximum possible mean life of state u, i.e., the mean life if the only mechanism of deactivation is spontaneous emission to state 1. The actual mean life T may be shorter than 70 if the quantum yield of the luminescence is less than one. VOLUME 37, NUMBER 4 In the derivation of Eq. (1), it is necessary to assume that the absorption band is sharp, and that the fluorescence occurs at the same wavelength as the *This work was carried out under the auspices of the U. S. Atomic Energy Commission. AUGUST 15, 1962 ¹ A. Einstein, Physik. Z. 18, 121 (1917). 2 R. Ladenburg, Z. Physik 4, 451 (1921). 3 R. C. Tolman, Phys. Rev. 23, 693 (1924). 4 (a) F. Perrin, J. phys. radium 7, 390 (1926); (b) G. N. Lewis and M. Kasha, J. Am. Chem. Soc. 67, 994 (1945). It is the purpose of this paper to present a modifica- tion of Eq. (1) which is applicable to polyatomic molecules under certain conditions. We shall first derive the equation, and in doing so, for the sake of completeness, we shall repeat the essential points of the derivation of the earlier formulas. We shall then summarize calculations of the fluorescent lifetimes of a number of molecules, and describe the experiments by which we measured these lifetimes. Finally, we shall discuss the limitations of the formula and the cases where it may not be valid. DERIVATION OF EQUATION FOR MOLECULES Consider two electronic states of a molecule, a ground state and an upper state u. The corresponding electronic wave functions may be called 0, and 0₂. 5 E. J. Bowen and W. S. Metcalf, Proc. Roy. Soc. (London) A206, 437 (1951). 6T. Förster, Fluoreszenz Organischer Verbindungen (Vanden- hoeck & Ruprecht, Göttingen, 1951), p. 158; F. E. Stafford, Ph.D. dissertation, University of California, Berkeley, 1959, UCRL-8854; V. N. Soshnikov, Soviet Phys.-Uspekhi 4, 425 (1961). 814 04 January 2024 10:42:39 FLUORESCENCE LIFETIME OF MOLECULES Associated with each electronic level are a series of states having different wave functions for nuclear motion, vibration, rotation, and translation. For the present purpose it is sufficient to consider only the vibrational states, though the rotational and transla- tional motion might be treated in an analogous manner. In the Born-Oppenheimer approximation, each vibronic state has a wave function V which can be written as a product of an electronic function and a vibrational function Þ. Va-OtÞla, ¥ub=OuÞPub. For simplicity in discussion, it will be assumed that each state is single; if degeneracies occur in either the electronic or vibra- tional part, they can be summed over in the proper manner when necessary. Relationship Between Einstein A and B Coefficients Suppose a large number of these molecules, immersed in a nonabsorbing medium of refractive index n, to be in thermal equilibrium within a cavity in some material at temperature T. The radiation density (erg cm-³ per unit frequency range) within the medium is given by Planck's blackbody radiation law p(v) = (8πhv³n³/c³) [exp(hv/kT)−1]¹. (2) By the definition of the Einstein transition proba- bility coefficients, the rate of molecules going from state la to state ub by absorption of radiation is NaBlaubp (vla-rub), (3) where N is the number of molecules in state la, and Vlaub is the frequency of the transition. Molecules in state ub can go to state la by spontaneous emission with probability Aubla or by induced emission with probability Bublap (Vub→la). The rate at which molecules undergo this downward transition is given by Nub[Aub+la+BublaP (Vubla)], (4) where Bubla-Blaub and Vubla=Vlaub. At equilibrium the two rates must be equal, so by equating expressions (3) and (4), it is found that Aub-la/Bub-la b→la=[(Nta/Nub)−1]p(vubla). (5) According to the Boltzmann distribution law, the numbers of molecules in the two states at equilibrium are related by Nub/Nta= exp[-(hvub->la/kT)]. la (6) Substitution of Eqs. (2) and (6) into (5) results in Einstein's relation Aub la=8rhvubla³n³c³Bubla (7) Relation of B Coefficients to Absorption Intensity In a common type of absorption meaurement, a beam of essentially parallel light is passed through the sample contained in a cell having planar windows perpendicular to the light beam. It may be assumed for convenience that the light beam has a cross section 1 cm² with a uniform intensity over this area. If p(v, x) is the radiation density in the light beam after it has passed a distance x centimeters through the sample, the molar extinction coefficient e can be defined by p(v, x)/p(v, 0) = 10-e(v)Cx=2.303e(v) Cz > where C is the concentration in moles per liter. If a short distance dx is considered, the change in radiation density may be written −dp(v)=2.303€(v)p(v, 0) Cdx. (8) For simplicity, all the molecules will be assumed to be in the ground vibronic state, Vzo. (This may not be true at normal temperatures, but this will not materially affect the results. It would be possible, but more complicated, to carry through a number of states with appropriate Boltzmann weighting factors.) It is easily seen that Cdx=1000N 0N-¹. (9) Furthermore AN(v), the number of molecules excited per second with energy hv, is given by (10) AN (v) = −cdp(v)/(hvn). Combining Eqs. (8), (9), and (10), it is found that AN (v)/Nto=[2303ce(v) /hvnN]p(v, 0). (11) Equation (11) gives the probability that a molecule in state 10 will absorb a quantum of energy hv and go to some excited state. To obtain the probability of going to the state ub, it must be realized that this can occur with a finite range of frequencies, and Eq. (11) must be integrated over this range. p(v, 0) can be assumed constant over this range and equal to p(vwo-ub). The value of e(v) must be only that for the one vibronic transition; if the spectrum is well resolved, this would present little difficulty, but it can be done in principle in any case. Then 815 (12) AN 10-ub N 10 This equation shows that, for a molecule in state 10 in a beam of parallel light, the probability of undergoing a transition to state ub is proportional to p(v10+ub), the constant of proportionality being the term in brackets. Expression (3) gave a similar relation for a molecule in an isotropic radiation field, except with proportionality constant Bob. In the latter case, the photons might be thought of as arriving at a molecule from random directions, whereas in the first case they all arrive from one direction. If, however, the mole- cules are randomly oriented, the average probability of absorption for a large number of molecules must be the same in either case for the same total radiation density. Therefore, 2303c hn [e (v)d Inv]o (vious). Bio-sub = 2303c hn N fed Inv, (13) 04 January 2024 10:42:39 816 where the integration is over the 10-ub vibronic absorption band. Still assuming all the molecules to originate in the state 10, it is possible to sum expression (13) over all the vibrational levels of the upper electronic state, to obtain a probability coefficient for all transitions to the electronic state u. This is given by 2303c hnN S. J. STRICKLER AND R. A. BERG Βιο-ι= Σβιο =ΣB10-+ub= (14) where the integration is now extended over the whole electronic absorption band of the l→→→u transition. fed fferat ed Inv, (15) The function 0₁(x, Q) is an electronic wave function for nuclei fixed in some configuration, so it contains the nuclear coordinates as parameters. If M(x) is the electric dipole operator for the electrons, it is well known that the probability for induced absorption or emission between two states is proportional to the square of the matrix element of M(x) between the two states. The constant of proportionality is of no importance here, so it will be designated as K. Blarub = Bub-rla where Lifetime Relationship for Molecules When a molecule undergoes an electronic absorption process, it will often end up in some excited vibrational level of the upper electronic state. However, if it is in a condensed medium, it will usually lose energy by collision until it is in the lowest vibrational level of the state. In fluorescing, the molecule may then go to various vibrational levels of the ground electronic Taking the appropriate sums, we find that state. Thus in absorption we observe the transitions 10-sub, while in fluorescence we observe 40-da. B₁o-=[Bw.us=K | M₁ (0) ³ Σ | [ ¹₁.*ÞadQ |*, Φια 20-ub tu b b It is therefore necessary to find a relation between B10»u=ΣòB10»uò and Auo»1= Σa^uo»la. B₁0u = K | M₁u (0) |², The wave functions of vibronic states are functions of both the electronic coordinates x and the nuclear coordinates, which can be taken as normal coordinates Q, since we are neglecting rotations and translations. They can be written as products of electronic and vibrational parts, for example, Vra (x, Q)=0₁(x, Q) Þra(Q). Using (15), the integral in this expression can be written Vra*(x, Q)M(x)¥ub (x, Q) dxdQ is an electronic transition moment integral for the transition, assuming the nuclei to be fixed in a position Q. It can be expanded in a power series in the normal coordinates of the molecule: M₁u(Q) =M₁u (0) +Σ(ƏM₁u/ƏQr) oQr+…….. (17) T For strongly allowed transitions in a molecule, and for reasonably small displacements from the equi- librium nuclear configuration, the zeroth-order term in this expansion should be the dominant one. It is only for transitions which are forbidden by symmetry or are weak for other reasons that zeroth-order term is small and higher-order terms become important. Let us make the assumption that we are dealing with a strong, allowed transition, so that only the zeroth-order term in (17) is important. Then Eq. (16) reduces to M₁(Q) = [0,*(x, Q)M(x)0u(x, Q) dx Bta-ub=Bub-ta=K | Mt« (0) |ª | √ $1a*¤«rdQ. (18) la la (19) since the Pub comprise a complete orthonormal set in Q space. The quantity necessary for calculation of the life- time is Ao, the rate constant for emission from the lowest vibrational level of electronic state u to all vibrational levels of state 1. Using Eqs. (7) and (18) this can be written Auo+1=[Auola=(8πhn³/c³) K | M₁(0) |² a =K |fferra ¥₁a*(x, Q)M(x)Vw(x, Q)dxdQ (16) by dividing by Σ S✨₁a*ÞuodQ |² = 1: • ΣPub=le² | [P₁a*PuidQ* (20) Φια a It is desirable to be able to evaluate the term Σa³u0→la³ | SÞ1a*ÞudQ |² experimentally. If the fluo- rescence band is narrow, ³ can be considered a constant and removed from the summation, the remaining sum being equal to unity. A better procedure can be derived [³u0-»la² | f®1₁a*PuodQ a Σ fi "Bude a = [Þra*(Q)M₁ (0) Þwo (0)dQ, Each term in the numerator of this expression is Φια lu ub proportional to the intensity of one vibronic band in the fluorescence spectrum. Each term in the de- nominator is proportional to v-³ times the intensity of one vibronic band. The sums over all vibronic bands can be replaced by integrals over the fluorescence 04 January 2024 10:42:39 FLUORESCENCE LIFETIME OF MOLECULES spectrum, so that the expression reduces to [1(v) dv Auo+1= fr-³1 (v) dv the reciprocal of the mean value of 3 in the fluo- rescence spectrum. This can be obtained experimentally. It should be noted that I(v), the intensity in the spectrum, must be measured in terms of relative numbers of quanta at each frequency, rather than in the usual energy units. Now, by combining Eqs. (14), (19), and (20), we obtain (v₂-³) Av-1₂ 8X2303Tn² c²№ ۱۱۰ مواره مورد fed Inv. (21) If either or both of the electronic states are degenerate, it is necessary to sum over the possible transitions. If it is assumed that all the possible transitions between degenerate components are equally allowed (which may or may not be true), or that there is a rapid equilibrium established between molecules in the dif- ferent component states of each degenerate level (which is probably true under normal conditions), the result is a factor of g₁/gu on the right-hand side of Eq. (21). It is convenient to write this equation in terms of the more common units where frequency is measured in cm¹ rather than sec-¹. The result is 1/70A₂0+1=8×2303mcn²N−¹(55-³) Av -18fed Ini = 2.880×10-³n² (ỹ,³¹).„._-¹³¹ √ed Inī, (22) where the integral is over the whole of the electronic absorption band concerned. This is the desired formula, applicable to strong transitions in molecules. If, as is the case for most atomic transitions, the band is sharp and the absorption and fluorescence occur at the same wavelength, ỹ can be considered a constant. A factor of 1/ can be removed from under the integral sign, and (ỹƒ¯³) Av-¹ becomes just ³. In that case, Eq. (22) reduces to Eq. (1), as must be expected for atomic transitions. COMPOUNDS FOR TESTING LIFETIME RELATIONS In choosing molecules for testing Eq. (22), we were guided by several considerations. (a) The lowest energy singlet-singlet electronic absorption band of the molecule must be fairly strong, as this was assumed in the derivation. (b) This first band should be well separated from other absorption bands so that the area under the experimental curve can be measured accu- rately. (c) Absorption must occur at wavelengths of 817 3650 Å or longer, as our equipment has glass optics. (d) The quantum yield of fluorescence must be known. In addition, to avoid complicating assumptions about the mechanism of quenching, we used only compounds whose fluorescence yields in solution at room tempera- ture were quite high, preferably nearly equal to one. On the basis of these criteria, we chose seven com- pounds for testing the formula derived in the preceding section. The names, structures, and absorption and fluorescence spectra of these compounds are shown in Fig. 1. The absorption intensity scales vary for dif- ferent molecules. The fluorescence spectra are in units of relative numbers of quanta, except for rhodamine B and rubrene where the scale is plate blackening. N-methylacridinium chloride was made by reacting acridine with dimethyl sulfate in benzene solution. An aqueous solution of the resulting N-methylacridinium methyl sulfate was treated with saturated NaCl solution to precipitate the chloride. The other com- pounds were commercially available materials. All compounds were purified by recrystallization. The absorption spectra were measured on a Cary model 14 spectrophotometer. A few of them require comment. 9-aminoacridine apparently exists as a neutral molecule in ethanol solution. However, in ethanol with HCl, or in water with or without HCl, the spectrum has a somewhat different appearance, probably indicating the presence of a protonated ion. Rhodamine B also exists as an equilibrium mixture of the structure shown and a colorless isomer having a lactone structure. The amount of the colorless form present in ethanol solution is not known, but our final results indicate that it is rather slight. In acid ethanol, rhodamine B forms a protonated ion whose spectrum is similar, but somewhat more intense and shifted to lower energies. As shown in Fig. 1, the first band in N-methylacridinium chloride is overlapped considerably by the strong second transition. This made it necessary to extrapolate the high-energy side of the first band in a rather arbitrary fashion, as indicated by the dotted curve. Because of this, the integrated area under the curve could not be measured with accuracy. The other molecules have much less uncertainty from this source. The fluorescence spectra of most of the molecules were measured on an American Instrument Company spectrophotofluorometer. It was felt necessary to make some correction for instrument sensitivity, although extreme accuracy was not necessary. For this purpose we used correction factors published by White, Ho, and Weimer, even though these are not strictly applicable to our instrument. To obtain (ỹ³)¯¹, two curves were plotted for each spectrum, one of the corrected intensities in units of relative numbers of quanta at each frequency, and the other of 3 times this value. 7 S. Wawzonek, Heterocyclic Compounds, edited by R. C. Elder- field (John Wiley & Sons, Inc., New York, 1951), Vol. II, Chap. 13. & C. E. White, M. Ho, and E. Q. Weimer, Anal. Chem. 32, 438 (1960). 04 January 2024 10:42:39