What the Einstein Relations Tell Us 1. The rate of spontaneous emission A21 is proportional to υ 3. At higher frequencies A21 >> B(υ) and all emission is spontaneous. A 21 = 8π hν3 c 3 B(ν) 2. Although they came from different points of view (one kinetic looking at the rate of change of numbers of molecules and the other from attenuation in intensity at steady-state). The Einstein coefficient B(υ) for stimulated absorption must be related to the molar extinction coefficient, ε(υ) in the Beer-Lambert Law at any given wavelength or frequency. 3. The rate of spontaneous emission A21 is related to the constant of stimulated absorption (absorption) and stimulated emission. Stronger something absorbs the stronger will be the emission.
Jablonski Diagram Singlet State Manifold Internal Conversion Intersystem Crossing Triplet State depicts various radiative and nonradiative processes that can take place between electronic states. Ground Electronic State, S0 Red-shifted fluorescence Red-shifted phosphorescence Absorption of radiation
Intrinsic Fluorescence Lifetime ln N 2 Short pulse of light N 2 time How the intensity of fluorescence decreases over time (when excitation stops) due only to spontaneous emission (it assumes stimulated emission is neglible). y = mx + b equation of a straight line By definition we equate the rate of fluorescence to an intrinsic lifetime τ. Nt d[n 2 ] dt d[n 2 ] [N 2 ] d[n 2 ] N 0 [N 2 ] = A 21 [N 2 ] = A 21 dt = t t=0 ln N t N 0 = A 21 t N 2 (t) = N 0 e ( A 21t) A 21 = 1 τ A 21 dt time Exponential Decay of Excited State N 2 (t) = N 0 e t/τ
N 2 Fluorescence Lifetime time Exponential Decay of Excited State N 2 (t) = N 0 e t/τ Points: Because there are other non-radiative pathways (internal conversion, intersystem crossing, quenching, etc.) for to rid molecules from the excited state 1/τ = A 21 is for molecules that show only deexcitation via fluorescence. Since the intensity of radiation is proportional to the number of excited state molecules per unit volume thus we can transform our equation to one that is experimentally useful. I(t) A 21 = k f = rate of fluorescence = 1/τ Emitted Intensity of Fluorescence I(t) True Decay time Exponential Decay of Excited State I(t) = I 0 e k f t = I 0 e t/τ
Non-Radiative Processes That Impact S1 Population Levels and Fluorescence. 1. Internal Conversion: excitation energy in S1 is lost by collision of excited state molecules with the solvent or by dissipation of energy as heat through vibrational modes in the excited to ground state. 2. Quenching: De-excitation resulting from collisions with certain types of molecules that can quench or remove energy from the excited singlet state. 3. Intersystem Crossing: small degree of forbidden spin exchange converts excited state singlet to triplet state which can convert to ground state by phosphoresance or by internal conversion. All of these processes serve to remove energy from the excited singlet state and therefore will decrease fluorescence intensity and quantum yield.
Rate Equations To Describe Fluorescence Process Reaction Rate Excitation S0 + hυ S1 Ia Fluorescence S1 S0 + hυ A21[S1] Quenching S1 + Q S0 + Q kq[s1][q] Internal Conversion Intersystem Crossing S1 S0 S1 T1 kic[s1] kisc[s1] Quenching, internal conversion and ISC are non-radiative pathways. Now let s analyze how the excited state population, S2 changes under conditions of steady state----constant illumination. We will sum the rates up and down and relate it to the Fluorescence intensity, IFluorescence.
Non-Radiative Processes And Fluorescence From the Einstein equations we can write the intensity of fluorescence as: I F luorescence = A 10 [S 1 ] d[s 1 ] dt = 0 = I a A 10 [S 1 ] k ic [S 1 ] k isc [S 1 ] k q [Q][S 1 ] = I a (A 10 [S 1 ] + k ic [S 1 ] + k isc [S 1 ] + k q [Q][S 1 ]) = I a [S 1 ](A 10 + k ic + k isc + k q [Q]) = I a [S 1] τ F I is the intensity of fluorescence, A 10 is the spontaneous emission coefficient (the rate constant if you will) and [S 1 ] is concentration of excited state molecules. Now we write the rate equation for changes in concentration of excited state molecules, S 1 under steady-state conditions (constant illumination let s say). Let s define: 1 τ F = A 10 + k ic + k isc + k q [Q] Now solve for [S1] and insert it into our original expression: IFluorescence = A10 [S1] I F luorescence = A 10 I a τ F = A 10 I A A 10 + k ic + k isc + k q [Q]
Quantum Yield = ΦF It is useful to define the a quantity which is the ratio of the number of photons emitted by fluorescence divided by the total number of photons absorbed to generate the excited state. It is called the quantum yield I F luorescence = A 10 I a τ F = A 10 I A A 10 + k ic + k isc + k q [Q] Φ F = I F I a = photons emitted photons absorbed = A 10 A 10 + k ic + k isc + k q [Q] = τ F τ 0 The quantum yield describes how efficient the radiative pathway is (or how the non-radiative pathways may dominate. If there are no nonradiative pathways then the quantum can be = 1 (rare if ever).
Instruments
Experimental Measurement of Fluorescence
The Fluorescence Spectrum With few exceptions, the fluorescence excitation spectrum of a single fluorophore species in dilute solution is identical to its absorption spectrum. Under the same conditions, the fluorescence emission spectrum is independent of the excitation wavelength, because all fluorescence occurs from the v = 0 vibrational level of the excited state. The Stokes shift is the gap between the maximum of the first absorption band and the maximum of the fluorescence spectrum Fluorescence Intensity Fluorescein molecule Stokes Shift is 25 nm 495 nm 520 nm Wavelength