Chapter 15 Lasers, Laser Spectroscopy, and Photochemistry

Transcription

1 Chaper 15 Lasers, Laser Specroscopy, and Phoochemisry ackground: In his chaper we will alk abou ligh amplificaion by simulaed emission of radiaion (LASER), heir impac on specroscopy and ligh-iniiaed reacions or phoochemisry. * Relaxaion Processes afer Elecronic Exciaion - once a molecule is excied i will no remain in ha sae forever evenually i will relax down o he ground sae - for simpliciy our sysem will be using a diaomic sysem -- below is a poenial energy plo in which we excie o a single excied sae, S1, which is in close proximiy o a lower placed riple sae, T1 (recall Hund s rules we would expec he riple sae o be lower in E han he single one) Figure 15.1 from ex -- according o our plo he equilibrium bond lengh, Re in increasing order is: Re(S0) < Re(S1) < Re(T1) -- line 1 in he figure is our absorpion energy -- also, we have vibronic ransiions in accordance wih he Frank-Condon Principle -- here are several differen mechanisms by which a molecule can relax from an excied sae --- radiaive ransiions: occur hru he absorpion/emission of radiaion --- noradiaive ransiions do no undergo absorpion/emission of radiaion -- back o he figure --- solid arrows are radiaive ransiions --- wavy arrows indicae nonradiaive ransiions w/in a single elecronic sae --- dashed lines-open arrow heads are nonradiaive ransiions bwn wo elecron saes -- for isolaed molecules: only process which conserve energy are allowed herefore in order o reurn o he ground a phoon mus be emied

2 -- when collisions bwn molecules occur: energy may be ransferred and lead o vibraional relaxaion (he wavy lines in our figure) --- due his relaxaion he molecule may quickly go down o he lowes vibronic sae in he excied sae, S1 A his poin he molecule can reurn o he ground sae hru wo mechanisms ---- eiher hru a phoon emission, line his ype of decay in which a ransiion occurs bwn 2 saes wih he same mulipliciy i is mos commonly known as fluorescence ---- or by nonradiaive decay, line once again his is a ransiion which occurs bwn 2 saes wih he same mulipliciy i is referred o as inernal conversion -- when wo excied saes of differen mulipliciies overlap he sysem can a nonradiaive ransiion bwn spin saes, line his is called inersysem crossing --- i requires a spin change in one of he e- s and so i occurs more slowly han inernal conversion ype nonradiaive ransiions --- once again relaxaion bwn vibronic saes can occur nonradiaively, he wavy lines in he T1 sae A his poin here are again 2 differen mechanisms o reurn o S eiher hru a phoon emission, line unlike line 2 his decay akes place bwn saes of 2 differen mulipliciies his is referred o as phosphorescence due o he spin sae change his ype of decay is slower han our friend fluorescence ---- or hru nonradiaive decay, line his is anoher case of inersysem crossing Summary of Transiions Process Transiion mulipliciy Time Scale Fluorescence Radiaive, S1 S s Inernal Conversion Collisional, S1 S s Vibronic relaxaion Collisional s Inersysem crossing S1 T s Phosphorescence T1 S s Inersysem crossing T1 S s -- Now, how does his ranslae o wha we would see in he specrum --- le Re(S0) = Re(S1) such ha he wo Morse poenials are aligned --- since vibronic relaxaion occurs much more rapidly han elecronic ones we can reasonably assume ha he excied molecule relaxes o he lowes vibronic sae before i fluoresces o he ground sae

3 Figure 15.2 from ex --- so, our absorpion specrum consiss of lines from he v = 0 level o he v = 0, 1, 2,... and are a higher E han he v = 0 v = 0 ransiion --- he fluorescence specrum consiss of lines from v = 0 v = 0, 1, 2,... and are a lower E han he v = 0 v = 0 ransiion --- also, boh he fluorescence and absorpion specra will conain ransiions bwn v = 0 and v = 0 levels --- he spacing bwn he lines in he fluorescence specrum depends upon he gaps bwn vibronic levels in he ground sae --- he opposie is he case for he absorpion specrum, he spacing depends on he gaps bwn vibronic levels in he excied sae --- if he vibronic frequencies are he same in boh he ground and excied saes hen he absorpion and fluorescence bands appear o be mirror images of each oher * Transiions beween elecronic saes & rae equaions - we know ha energy bwn wo elecronic saes in quanized - consider a sysem of Naoms which has a ground sae and one excied sae for simpliciy -- le N1 and N2 be he number aoms in he ground & excied saes such ha Noal = Naoms = N1 + N2 -- as one migh suspec he occupaion of each sae is emperaure dependen we will explore his concep nex semeser --- his emperaure dependence is on he order of kt where k is he olzmann consan --- if E2 E1 >> kt, he aoms do no possess enough energy o occupy he excied sae and he majoriy of hem are in he g.s., Noal = Naoms = N1 --- when radiaion, h12 his he sample on he order of E2 E1 hen some aoms will have enough energy o be promoed o he e.s. -- he energy densiy of he ligh is described by 2 quaniies --- radian energy densiy, = radian energy/uni volume (J/m 3 ) --- specral radian energy densiy, = d/d (J s/ m 3 )

4 --- herefore, we are mos ineresed in (12) -- Einsein proposed: dn1() dn1() rae pvv12 N1() or rae 12 pvv12 N1() d d where 12 is a proporionaliy consan an Einsein coeff --- in words: he rae of exciaion from he g.s. o e.s. is proporional o (12) and o N1() he number of aoms in he g.s. a ime --- he (-) occurs because N1() decreases wih ime -- for absorpion: dn1() dn2() 12 pv v12 N1() d d -- as we know aoms will no remain in his excied indefiniely and so Einsein proposed wo pahways by which hey could reurn o he g.s. --- sponaneous emission: aoms emi a phoon of energy, h12 = E2 E1 Figure 15.4 from ex dn2() ---- rae N2() A21N 2() d where A21 is anoher Einsein proporionaliy consan -- simulaed emission: exposure of he sysem o anoher phoon of energy, h12 will simulae he aom o reurn o he g.s. Figure 15.5 from ex dn2 () ---- rae 21v v12 N2() d --- noe: his ype of emission amplifies ligh inensiy since one aom is simulaed o emi anoher and hereby generaing anoher phoon, h12 -- when we expose our sample o ligh all hree processes occur; absorpion, sponaneous and simulaed emission dn1() dn2() 12 pvv12 N1() A21N2() 21vv12 N2() d d - how are our coefficiens relaed o each oher? -- le our sysem reach eq dn1() dn2() 0 d d

5 --- (12) is he eq specral radian energy densiy which we assume comes from a hermal blackbody radiaion source --- recall Planck s blackbody disribuion law: 3 8 h v12 v v12 3 hv12 c kt e we can ge a relaionship bwn (12) and he Einsein coeffs 12 pvv12 N1() A21 N2() 21vv12 N2() 0 A21N2 A21 pvv12 12 N1 21N2 A21N2 pvv12 N N N1 N nex semeser we will learn ha a sysem in eq a emperaure, T, he of number of aoms in he sae j wih energy Ej is given by E j kt N j ce where c is a proporionaliy cons --- we can use his relaionship o ge our N2/N1 raio hv12 N2 E2E1 kt kt e e N1 --- now we can plug his back ino our expression for (12) A21 A21 pv v12 N hv 1 12 N kt 12e going back o our Planck blackbody disribuion law, his expression is equivalen o our mos he above expression only when 12 = h v12 and A c 3 3 8hv12 8hv A c c 8 h v pv v hv12 hv12 hv 3 hv kt kt kt c kt 12e 21 12e e 1 e * A 2-Level sysem canno achieve a populaion inversion - lasers amplify ligh by simulaed emission - for his o occur he applied phoon mus simulae emission from an excied sae raher han be absorbed by he molecules in he ground sae leading o an increase in he excied sae populaion insead of ground sae populaion 21vv12 N2() 12 pvv12 N1() -- since 21 = 12 his means ha simulaed emission is more probable when N2 > N1 -- his siuaion is called a populaion inversion since our excied sae is more populaed o han he ground sae hv12 N2 E2E1 kt kt -- according o e e N2 mus be less han N1 since N1 hv12 0 herefore, his populaion inversion mus be a noneq kt 2

6 phenomenom - le s look a our wo-level sysem dn1() dn2() pv v12 N1() N2() AN2() d d -- if we assume a = 0 all he molecules are in he ground sae N1 = Noal and N2 = 0 dn2() pv v12 Noal () AN2() d pvv12 Noal A2pv v12 his has a soluion of N2 1 e A 2p v v12 -- if we plo N2 / Noal versus ime we will find ha even a infinie ime our excied sae populaion never ges above 1 which is wha we require o ge a populaion inversion N2 pv v 12 Noal A 2pv v in fac he whole expression converges o no quie ½: since A > 0 N2 N2 1 Noal N1 N herefore, we can never ge a populaion inversion for a wo-level sysem * Populaion Inversion and 3-Level sysems - we will once again assume ha each level is nondegenerae and represen a single sae of he sysem such ha E1 < E1 < E3 for saes 1, 2, 3 respecively - our mission is o show ha we obain a populaion inversion bwn saes 2 and 3 such ha we will generae lasing Figure 15.8 from ex -- he double-headed arrows are indicaive of boh absorpion and simulaed emission bwn he wo saes -- we will also assume ha ij = ji for any wo saes i and j -- a = 0 all he molecules are in he g.s. or Noal = N1() -- now, we apply a phoon of radiaion which will excie molecules from he

7 ground o he 2 nd excied sae or hv E E wih v v his ype of beam is ofen referred o as a pump source --- i is assumed ha no molecules/aoms are excied o he 1 s e.s. -- decay a his poin may occur eiher sponaneously o saes 1 or 2 or by simulaed emission o sae 1 -- for hose molecules which sponaneously emi o sae 2 hey can furher furher relax all he way down o sae 1 -- if phoon of radiaion is available wih frequency, 32, we can cause simulaed emission bwn saes 2 and his will be available since we will have sponaneous emission bwn saes 3 and 2 -- a his poin regardless of exacly which sae he aoms occupy: Noal N1 N2 N3 dn1 dn2 dn3 -- once he sysem reaches eq: d d d -- we can solve each sae explicily bu we will consider only sae being in he middle is fun! --- he populaion of sae 2 is dependen on: ---- sponaneous emission from 3 2, A23N sponaneous emission from 2 1, A21N simulaed emission from 3 2, (32)32N absorpion from 2 3, (32)32N2 --- a eq: dn2 0 A32N3 A21N2 vv32 32N3 vv32 32N2 d A N v N A N v N v v v v 32 3 v N A v N A v N A v 3 N A v v now, we can ge a populaion inversion if A21 > A physically, his occurs when sae 3 sponaneously decays o sae 2 more slowly han sae 2 sponaneously decays o sae his ype of sysem is referred o as a gain medium * Applicaions of Lasers - Resoluion: how well we can disinguish bwn absorpion/emission peaks -- specral resoluion, in paricular, is he limi of a specromeer --- in a lamp-based insrumen his is around 0.03 cm he monochromaic ligh generaed by a laser in he range of visible has a specral widh of 3.5 x 10 cm -1 - Phoochemical Dynamics -- recall a phoochemical process is one ha is iniiaed by ligh absorpion -- some of hese process are see ex for prey examples (p. 615)

8 --- phoodissociaion --- phooisomerizaion --- phoodimerizaion -- we judge he success of hese reacion hru he quanum yield, number of molecules ha undergo reacion number of phoons absorbed * How do we conrol geing differen ypes of decay? his was aken from: D. C. Harris and M. erolucci, Symmery and Specroscopy: An Inroducion o Vibraional and Elecronic Specroscopy, Dover, 1989, p he abiliy of a molecule o fluoresce or phosphoresce is dependen on he lifeimes (given in he char in he char previously) of hese wo process relaive o heir compeiors of inersysem crossing and inernal conversion - also we can increase or decrease hese processes by inroducing oher reagens or hru some oher environmenal conrol (e.g. emperaure) see p. 360 in he handou for a much more deailed explanaion -- for example in order for phosphorescence o occur we can faciliae his spinforbidden process hru he inroducion of a paramagneic/heavy aom --- if we incorporae O2 ino he sysem his will increase he phosphorescence we generae --- since phosphorescence and fluorescence are also compeiors, an increase in one will lead o a decrease in he oher -- in order o ruly generae phosphorescence we need o slow down all he oher processes which occur in he sysem --- if we lower he emperaure his will slow down all compeing processes --- i urns ou ha such a environmen no only leads o phosphorescence bu may even enhance i - one final noe: aside from emperaure and oher reagens, solven may also play an imporan role