Molecule in The Excited State - Optical Spectroscopy | PHYS 552, Study notes of Optics

Download Molecule in The Excited State - Optical Spectroscopy | PHYS 552 and more Study notes Optics in PDF only on Docsity! 1 Fluorescence lifetimes and how we measure them Consider only one molecular species in the ground state and in the excited state. Each separate species will have only a single exponential decay time from the excited state. In the last lecture we showed that if the molecule is excited by a very short intense pulse of light, of very short extent (much shorter than the exponential decay time that the output of fluorescence photons will decay as an exponential. The lifetime is the reciprocal of the sum of all the rates of deactivation. To get a good idea of how this is understood, we have the following analogy. MOLECULE IN THE EXCITED STATE - some statistics - Simple case: one door in, and one door out, one molecule occupation, molecule has no memory EXCITED STATE ROOM X \ ™ , sets time t=0 x 1 IN ke (a constant!) is the probability per unit time that X,*will exit through door F probability that X,*is still in the excited-state room at time t is: Every molecule exiting through the door "F" loses (emits) a photon hn,, # X,* molecules exiting per unit time through door F = [#X,*]k,; so d[#X,*]/dt = -[#X,*]k, - d[#X,*]/dt = d[#hn,,]/dt, so THIS IS THE INTENSITY OF FLUORESCENCE LIGHT ~ Now TWO EXIT DOORS TO EXIT (F AND T) F = FLUORESCENCE; T = TRANSFER erty _7 energy *+ ,/ acceptor i . Y;* one doorin, ‘' 1 two doors out, *, P one molecule .? vos | probability per unit time that X,* will exit through eithpr door F or door T is : k, + ke (a constant!) acme one door in Cesena ener two doors out, aenntet 2? (9 multiple molecules { * ‘ Y, "nt" ("n0") molecules in__»> excited-state room a en erg  10 QUANTUM YIELD OF FLUORESCENCE AND QUANTUM YIELD OF ENERGY TRANSFER If we integrate the total rate of leaving the excited state through all doors over all time, we should get the total number of molecules originally excited; i.e. [#X0*]. ( ) ( ) ( ) [ ] *][#1 )( *][# *][#*][# 0 11)(0 0 )( 00 11 Xe kk Xkk dteXkkdtXkk tastkk FT TF t tkk TF t TF FT FT ⎯⎯⎯ →⎯− + ⋅+ = ⋅+=⋅+ ∞⇒+− +−∫∫ ; so, this is OK ----------------------------------------------------------------------------------------------------------- If we integrate the rate of leaving the excited state through only the “F” doors over all time, we should get the total number of photons emitted. [ ] *][# )( 1 )( *][# *][#*][# 0 11)(0 0 )( 00 11 X kk ke kk Xk dteXkdtXk FT Ftastkk FT F t tkk F t F FT FT + ⎯⎯⎯ →⎯− + ⋅ = ⋅=⋅ ∞⇒+− +−∫∫ The quantum yield of fluorescence is the fraction of excited molecules that emit a photon; that is: Quantum Yield of Fluorescence = )(excited originally molecules ofnumber total emitted photons ofnumber total FT F kk k + = ------------------------------------------------------------------------------------------------------------ -------------------- If we integrate the rate of leaving the excited state through only the “T” doors over all time, we should get the total number of transfer events; i.e. [#Ytotal*] [ ] *][# )( 1 )( *][# *][#*][# 0 11)(0 0 )( 00 11 X kk ke kk Xk dteXkdtXk FT Ttastkk FT T t tkk T t T FT FT + ⎯⎯⎯ →⎯− + ⋅ = ⋅=⋅ ∞⇒+− +−∫∫ The quantum yield of energy transfer is the fraction of excited molecules that transfer a quantum of energy to the acceptor (Y); that is: Quantum Yield of energy transfer = )(excited originally molecules ofnumber total ed transferrquantaenergy ofnumber total FT T kk k + = 11 Measurement of the dynamics of the fluorescence decay The fluorescence phenomena that we are interested in right now are linear mathematical systems. So, we can use the casual superposition relationship: ( ) ( ) ( ) 0 ' ' ' t f t p t i t t dt= −∫ and we also have the following ( ) ( ) ( ) 0 '' '' '' t f t p t t i t dt= −∫ The fluorescence response function is i(t) and the exciting light function is p(t). The form of the integral equation is called a convolution integral. It is very a general equation in physics and engineering, and is used to describe many physical systems. The equality of the two equations is simply a general result of Fourier analysis and convolution theory. The different experimental methods for determining the form of the signal, depends on the form of p(t) – i.e. the type of excitation process that is used. These equations are general for all types of excitation. Direct time response measurement (time domain measurement): The response function for a delta function pulse ( ( )( )p t tδ= ) is ( ) f t f t e τ − = for a single fluorescent molecular species. The instrument response is never perfect. The instrument response function is I(t). This means if there were a delta function light pulse falling on the detection system, one would record a signal I(t). So, for a fluorescence signal i(t) we would measure: ( ) ( ) ( ) 0 ' ' ' t f t I t i t t dt= −∫ So, if the response is fast compared to the fluorescence decay, the measured signal will be essentially ( ) ( )f t i t= . If the response of the instrument is not fast, or if the light pulse is not short, compared to the fluorescence decay, or both, then we have to de-convolute the true fluorescence response (i(t)) from the recording of the fluorescence signal (f(t)). The form of the fluorescence decay, i(t) is in general a linear sum of exponentials with different lifetimes and with different exponential amplitudes. If flash lamps are used as excitation sources, the excitation pulse is generally several nanoseconds long, and this requires de-convolution of the measured signal to extract the actual fluorescence decay parameters. In our laboratories, we use two-photon excitation with Ti-Sapphire lasers that 12 have on the order of 100 femto-second pulses, and this is short enough to approximate as a delta function. Then the only de-convolution required is that of the instrument response function. The detectors are usually photomultipliers or diodes (or avalanche photodiodes). The usual technique is the “photon-sampling” technique. The pulsed excitation intensity is adjusted so that on the average only one photon is detected for every 20th or 100th excitation pulse. This is to ensure that only one photon is detected, and that there are not overlapping detected photons (photon pile-up). That is, we count photons. The signal of the photomultiplier (the pulse that is generated by a single detected photon and amplified by the dynode structure of the photomultiplier) is passed through a discriminator. The discriminator level is set so that only a pulse greater than a preset amount will be be recorded at the output of the discriminator. The level of the discriminator is set so that the dark noise of the photomultiplier (current fluctuations at the anode of the photomultiplier that is generated randomly by other means than a photon hitting ther cathode) is not detected. A fast clock is started by a second photomultiplier circuit that detects the excitation pulse, which exists of a relatively high flux of photons during the short pulse. This sets and defines the time zero. The clock runs until the detection circuitry receives a pulse from the detector photomultiplier (with the discriminator on the output); this signal records the arrival time of the detected photon at the photomultiplier cathode (plus the time it takes for the detected pulse to traverse the photomuliplier circuitry). This signal indicating a detected photon stops the clock; the time from the last excitation pulse to the time of the detected photon is recorded and read out to the computer. The clock is reset, and the process is repeated. The light pulse can be a flash lamp, a pulsed repetitive laser or sometimes pulsed light from a synchrotron. The time delays of the system, and the instrument response function, are determined using scattering of the excitation pulse as a control. As one might expect, extreme care is necessary to detect such fast signals, and there are many artifacts of the measurement that must be known, determined and corrected. For instance the photomultipliers have a color effect; the transit time of a pulse through the photomultiplier depends on the color of the photon hitting the cathode. This contributes to the error in the determination of the instrument characteristics when using scattering of the excitation pulse as a standard. Also the transit time depends on the position that the photon hits the cathode – this effect will contribute to the noise of the measurement. But in general it is possible to make extremely accurate determinations with this method provided that sufficient counts are acquired (this often requires over 105 counts per bin time (the shortest time of digital acquisition). This type of method for measuring fluorescence lifetimes has been used extensively, and there has been a tremendous amount of numerical research expended to develop robust and accurate methods for fitting the data. The methods for fitting the data are quite sophisticated. This is because the signal usually consists of more than one exponential, and exponentials are notoriously non-orthogonal – that is, there is no way analytically to separate the individual components of a sum of exponentials. If the fluorescence signal consists of a sum of exponential decays that are not well separated in time, it is very