1. Strahlungsgesetze, Ableitung der Planck-schen Strahlungsformel, Einstein-Koeffizienten, Extinktinskoeffizient, Oszillatorenstärke

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1 1. Strahlungsgesetze, Ableitung der Planck-schen Strahlungsformel, Einstein-Koeffizienten, Extinktinskoeffizient, Oszillatorenstärke Einheiten in diesem Kapitel: diesmal cgs. Energy volume density of blackbody iation (Planck s law) Energy volume density E/V [erg cm - ] per frequency [s -1 ] E V ( / ) 8 ρ( ) = πh c ( exp{ h / kt} 1) 1 [erg cm - s] (1.) or per wavenumber [cm -1 ] E V ) ( / ( ) 8 1 = hc ρ π υ ( exp{ hcυ / kt} ) 1 [erg cm - ]. (1.4) with k Boltzmann s constant.

2 Einstein coefficients A and B for a narrow molecular transition at Rate constant [s -1 ] for emissive transition 1 (or probability for transition in unit time) as 4 64π υ 8π w 1 M M ( ) hc + h ρ = or (1.6) w 1 = A + B ρ( ) with Einstein coefficient A [s -1 ] for spontaneous emission and B [erg -1 cm s - ] for induced absorption. M is the transition dipole moment e qˆ 1. Oscillator strength 8π m = M (1.7) he which may also be written hm = B (1.8) πe

3 Relationship between A and B using detailed balance The absorptive transition rate constant w 1 [s -1 ] between states and 1 is written w Bρ( ) B ρ( = = ) (1.9) 1 υ where B [erg -1 cm s - ] and B [erg -1 cm s -1 ] are the corresponding Einstein coefficients for absorption. B / c = B (1.1) Stationary equilibrium between upward absorption 1 and downward emission 1 : ( A ) exp{ E / kt} Bρ = exp{ E1 / kt} + Bρ (1.11) hence ( h / kt} 1) Bρ = A exp{ (1.1) Substituting ρ from eq. we obtain 8πhυ A B 8 hc = = π B (1.1) c

4 Coefficient B from absorption measurements of a narrow line at, and then M The ambert-beer law light intensity I after passage through a sample of length [cm]: I I e σ N = (1.14) where N [cm - ] is the particle density, σ [cm ] the cross section of a particle, and I is the impinging intensity. Chemists prefer the form I = I. (1.15) 1 ε m Here m is the molar concentration measured in [mole ltr -1 ] (by "chemical" exception to our cgs convention) and ε [ltr mole -1 cm -1 ] is the molar decadic extinction coefficient. Obviously the molar concentration m is m = 1 N / (1.16) N where N [mole -1 ] is Avogadro s number and factors 1 are always dimensioned [cm ltr -1 ]. It follows that 1 σ = ln(1) ε (1.17) N ln(1) is required because the exctinction coefficient is decadic.

5 Equivalently ambert-beer in differential form, for example for the energy volume density ρ [erg cm - ] which depends on spatial position x. d ρ(, x) = ρ(, x) ln(1) ε m dx. (1.18) Numbers of photons absorbed per unit time in a sheet of unit area [cm ] between x and x+dx at frequencies between and + d. The photon flux (photons per area propagating in direction x) which enters the sample at x is larger than the flux which leaves the sample at x+dx. The difference corresponds to a stationary excitation density, or photons absorbed per unit time in the sample volume: divide each side by h, and multiply by the velocity of light c, to get the change per unit time: dρ(, x) c d 1 c ε = ln(1) d ρ(,x) N dx (1.19) h hn For a narrow vibronic line: obtain the number of excitations per unit time, dn, in volume A dx where A is the unit area: c dn 1 ε = + ln(1) d ρ(, x) N dx hn (1.) Note the sign change on the right because for every photon absorbed, i.e. for energy density being reduced by h, an elementary excitation is counted positively. Hence the rate constant for absorption is 1 c ε w 1 = ln(1) d ρ(, x) hn (1.1)

6 Comparison with eq. 9 shows 1 c B = ln( 1) hn ε d so that (1.) 1 B = ln(1) hn ε d = ε d (1.) [erg -1 cm s -1 ] = [cm mole ltr -1 erg -1 s -1 ] [ltr mole -1 cm -1 ] Thus we obtain coefficient B from measurement of ε ( ). However it is far more useful to proceed to the transition dipole moment which becomes h c hcln(1) ε M = B = π π d = ε 8 8 N d (1.4) [esu cm ] = [esu cm ltr -1 mole] [ltr mole -1 cm -1 ] where esu g cm s -. Division by 1-6 gives M in Debye. [Debye= 1-18 esu cm]. The refractive index n of the solvent also plays a role, but for simplicity this is omitted here.

7 Coefficient A from emission lifetime measurements of a narrow line at, and then M So far the chosen vibronic transition was observed in absorption. Instead the iative lifetime τ may be measured in emission. For a single vibronic line one has from (1) k A hc 1/ τ = 8π B (1.5) which with () becomes 1 /τ = ε d (1.6) [s -1 ] = [cm 4 s -1 mole ltr -1 ] [cm - ] [ltr mole -1 cm -1 ] Note that the factor derives simply from the mode density of Planck oscillators at the emission frequency (cf. eq. ff). Eq. 5 will later be used to calculate the iative lifetime from broad absorption and fluorescence spectra. However here we are interested in the transition dipole moment and for that reason go back to the definition of A in eq. 6, to obtain immediately h k M = 4 64 (1.7) π [esu cm ] = [erg s] [s -1 cm ]

8 Bandshape function for absorption and fluorescence bands of distributed vibronic lines As was mentioned earlier, vibronic transitions for large molecules in solution are rarely resolved because of spectral congestion together with dynamic broadening due to solvent fluctuations. However the absorption spectrum ε ( ) is nonetheless well defined and composed of many overlapping vibronic lines. For example at some fixed frequency corresponding to the absorption frequency singled out in fig. 1, in fact several vibronic transitions i k could contribute to ε ( ) here and division by (see 1.4) should lead to M ik. In the simple Born-Oppenheimer (BO) approximation the ( el) ( nuc ) ik M M ik M = are factorized into a common electronic term, which is independent of frequency, and the nuclear Franck-Condon (FC) factors which are distributed over frequencies and integrate to one because of the completeness of the nuclear wave functions. Therefore we start from eq. 1.4 and define an absorption bandshape function as follows: 8π N ( el) = ( nuc) g ( ) ( ) / abs ε M M ik δ ([ k i ] ) (1.8) hcln(1) Here (...) g ( abs ) is an area-normalized distribution of FC factors for upward transitions over absorption frequencies and (exactly as in eq. 4) M ( el) hcln(1) ε = 8 d π N = ε d (1.9) In order to discuss the fluorescence spectrum let us start with the total fluorescence quantum yield Φ=k τ where τ is the excited-state lifetime. It will be more useful to consider the normalized yield 1 Φ = kτ being distributed over a large large spectral range because k is so distributed. We define the (area-normalized) distributions f and f Φ Φ f ( ), f ( ), and f (1/ ) f ( ) = (1.) Usually f () is measured since spectrometers or spectrographs are linear in wavelength; however for the moment we remain in the frequency representation. For example consider now the emission line singled out in fig. 1, which is observed at some emission frequency. Again several vibronic transitions k i could contribute to f ( ) here. Now division by (cf. eq. 7) gives the bandshape function for emission 4 π 64 ( el) = τ ( nuc) g ( ) ( ) / δ ([ ) f M M ki i k. (1.1) h Here (...) g ( ) is an area-normalized distribution of FC factors for downward transitions over emission frequencies.

9 Eq. 1 may be used to convert a fluorescence distribution f into an equivalent spectrum of the cross section for stimulated emission (SE). The idea is that, basically, the FC factors are distributed spectrally and that the distributions of ε or of k follow. For fluorescence one has from eq. 1 h 1 ( el) = ( nuc) f ( ) / δ ([ ) M M 4 ki i k (1.) 64π τ On the other hand for absorption along the same transitions and frequencies one has from eq. 8 hcln(1) ( el) = ( nuc) ε ( ) / M M δ ([ ] ) ik k i (1.) 8π N We obtain from comparison simply 1 σ ( ) ( ) / 8π τ f SE = (1.4) c In order to find τ one starts with eq. 1 where the factor describes how the cavity mode density rises with frequency. In case of distributed emission frequencies one should instead use the mean value ( ) d (1.) g where g ( ) is the normalized fluorescence bandshape function (...) in eq. 8. To obtain the iative lifetime one needs to modify eq. to obtain approximately 1 /τ ε [s -1 ] = [cm 4 s -1 mole ltr -1 ] [cm - ] [ltr mole -1 cm -1 ] (die sog. strahlende ebensdauer ) d (1.4)

10 The electronic transition dipole moment, according to eq. 9. M (el) is obtained from the absorption spectrum Oscillator strength for distributed vibronic absorption lines For a narrow line we found that the squared transition dipole can be determined experimentally from the absorption spectrum, as given by eq. 1.4: h c hcln(1) ε M = B = π π d 8 8 N On the other hand, one defines an oscillator strength as µ = [esu] (1.5) q where µ is the dipole moment [esu cm], q is the coordinate [cm] along which charge is being moved. Often the normalized oscillator strength = / e is given; it measures the fraction of elementary charge that is moved by the electric field. For optical transitions can be just below 1, while IR transitions have much smaller values (.1-.1). Theoretical considerations provide the relationship 8π m = M (1.6) he For a narrow absorption line it is seen, by combining 1.6 with 1.4, that ε, and more exactly = h m 1 c ln(1) ε πe h N d = ε d (1.7) [ ] = [ltr -1 mole cm ] [ltr mole -1 cm - ]

11 Für Spezialisten: Tabulation and conversion of absorption and fluorescence spectra Spectra are recorded at integral measurement wavelength points. Relative absorption spectra ε () [-] are normalized to 1 at the maximum in the first absorption band and the corresponding εmax [ltr mole -1 cm -1 ] is also stored. Fluorescence spectra f () [nm -1 ] are assumed to be area-normalized to 1: f ( ) d = 1 band When using the formulas which were derived in the previous sections for numerical work, remember that the use of the nm wavelength scale is properly considered. The refractive index n is best obtained by averaging over the emission band using f ( ) f ( ) n = n( ) d / d 5 5 representation representation formula in terms of ε () [ltr mole -1 cm -1,, all other units in cgs] and f () [cm -1 ] numerically given in terms of ε () [-], ε [ltr mole -1 cm -1 ], f () [nm -1 ], max τ [s] ε ε () ε ε ( ) f f () f () abs ln(1)1 σ ε ( ) N max ε max ε ( ) 1 1 f ( ) 17 1 f ( ) σ SE 4 4 8πc τ n τ n ε ε () ε ε ( ) ( ) f f abs ln(1)1 σ ε ε ( ) N max 1 1 f ( σ SE 4 8πc τ n ) max 7 f ( ) 1

12 An example for absorption and fluorescence measurements, and their analysis We examine the dye Coumarin 15 (C15) in -methyl-butane (mb): CF N O O Measurements: The absorption spectrum A() at a concentration conc [mol ltr -1 ] in cell with thickness d A( ) [cm] gives the extinction coefficient ε ( ) = [ltr mole -1 cm -1 ]: conc d r l t m ol e 1 c m 1 psilon e lambda nm This may be replotted against wavenumber (purple line below): r l t m ol e 1 c m 1 psilon e wavenumber cm 1 Actually only the transition S S1 is of interest, and this is estimated by removing (somehow) the overlaying second band which can be assigned to a higher electronic state, i.e. to the transition S Sn (n>1). The result i.e. the S S1 absorption band, is shown as a blue line. This is the ε( ) in the formulas (1.1-5), and it will be used in the analysis below.

13 The emission spectrum is usually recorded as fluorescence photon counts F ( ) over wavelength [nm -1 ]. The total number of photons counted is Φ = F ( ) d. If we would receive only one photon, then its probability distribution over wavelenghts would be f ( ) = F ( ) / Φ [1/nm -1 ]. We can think of f ( ) = Φ / where Φ is just one emitted photon. To sum up: we measure F ( ), determine its integral Φ, and form area-normalized f ( ) The latter is shown in the following figure:. n f qd o ve r w avelengt h 1 m lambda cm 1 Transformation of the recorded emission spectrum: The fluorescence quantum distribution must (for calculations) be given over wavenumbers. 7 We have 1 f ( ) = Φ / = ( Φ / )( / ) = f ( ) = f ( ) where is measured in 7 1 nm and in cm -1. In the following figure, the green line from above has simply been replotted against wavenumber cm 1 Obviously this alone does not create the desired fluorescence quantum distribution over wavenumbers in addition the green line must be divided by. The effect of that operation is: pushing the low-energy side up, bringing the high-energy side down, as shown by the solid red line for f ( ). But actually we want to go further and generate the lineshape

14 function g ( ) using 1.1, which gives (essentially) the dashed red line above. Note that in going from the solid green line to the dashed red line, we have divided by 5! After normalisation the distribution of oscillator strength g ( ) is obtained, and this is shown together with ( ) next: g abs 1 g 1 m c wavenumber cm 1 Remember that g ( ) ε( ) / abs and then normalized. Analysis From equ. 1.9 we get the transition dipole matrix element M=5.41 D, and from eq. (1.7) the oscillator strength =.56. The average (emission frequency) is = g ( ) d = 1479 cm -1 ; the average (emission frequency) is = g ( ) d = (1598 cm -1 ) The iative rate is calculated (using 1.4) to be k= s -1, and the iative lifetime becomes τ=1.8 ns. The observed lifetime is smaller because noniative processes (mainly IC in this case) also remove population from S1. Notice that the vibrational structure in the oscillator distributions g ( ) and g ( abs ) differs. This indicates geometrical change in S1 after excitation, before emission. We will come to this point later (Franck-Condon factors, vibrational progression).