10. Second quantization: molecule-radiation interaction

Transcription

1 10. Secod quatizatio: molecule-radiatio iteractio Now that the full molecular (sectios 7 through 9), eld (b), ad iteractio (b) Hamiltoia operators are i had, they ca be combied to yield the overall molecule-radiatio Hamiltoia. A quatum treatmet of the radiatio eld results i particles (photos) which ca be destroyed or createdby molecular absorptio ad emissio. The level of theory where matter particles are treated by a particle quatum Hamiltoia, ad eld particles are treated by a quatized eld Hamiltoia from sectio is referred to as "secod quatizatio." We will cosider this i detail here. The ext higher level of theory also treats the matter particles with a eld Hamiltoia, allowig them to be created or aihilated (e.g. electro-positro collisio), ad is referred to as "quatum electrodyamics." 10.1 The full quatized Hamiltoia Addig up the eld ad charge-eld eergies from sectio b to obtai the total Hamiltoia yields p H tot = + V (r) + ωa * m k, λ a k,λ q k,λ m c (p A + A p ) + q m c A 10-1 = H mol + H eld + H iteractio The quatum versio of H mol was derived i sectios 7 through 9, ad H eld is just a harmoic oscillator Hamiltoia. The oly problem for quatizatio is the potetial o-commutatio of p ˆ ad A ˆ. However, p ˆ A ˆ Ψ = i (AΨ) = {( A)Ψ + A Ψ} = A ˆ p ˆ Ψ 10- i if we use the Coulomb gauge A = 0. The rst iteractio term i eq simply becomes (1) q q H it = p A = { Vk} hc 1/ p e ˆ k, λ {a + k,λ e ik r + a k,λ e ik r } 10-3 mc mc,k,λ if A is expaded i plae waves as i eq. -7. Schrödiger represetatio, Ĥ = ˆp + V(r ) + ω m k,λ q m c,k,λ (â + k,λ â k,λ + 1 ) { hc Vk }1/ ˆp ê k,λ {â k,λ e ik r Quatizig the full Hamiltoia i the + â + k,λ e ik r } 10-4 q + { hc m c,k,k ',λ Vk }{â kλ â + k ' λ e i(k k ') r + â + kλ â k ' λ e i(k k ') r + â kλ â k ' λ e i(k+k ') r + â + kλ â + k ' λ e i(k+k ') r } where the rst lie icludes the ucoupled molecule ad eld, the secod lie icludes the rst order iteractio Hamiltoia (liear i A, where A has bee replaced by the expressio derived i terms of plae waves i sectio b), ad the third lie icludes the secod order iteractio operator (quadratic i A). As i sectio b, V is the volume of the box cotaiig the radiatio eld ad molecule, ad λ=1, are two polarizatios perpedicular to k. 10-1

2 I this picture, the radiatio eld appears like a collectio of harmoic oscillators coupled to the molecule via H (1) ad H () : H eld = ω k ( N ˆ +1 / ) 10-5 k, λ k, λ Each of the iitely may combiatios of wavevector k ad polarizatio λ correspods to oe harmoic oscillator-like eld mode, which ca be populated with N k,λ eld quata. These eld quata are referred to as photos. We eed to d a appropriate eld basis set for the ucoupled case, ad the most reasoable choice is just a product of harmoic oscillator fuctios, which is a eigefuctio of the eld Hamiltoia. H eld N kλ > N k' λ ' > = H eld N kλ >= ω( N kλ + 1 ) N kλ > The operators a kλ ad a kλ have the usual effect o the eld eigestates: a + kλ N kλ >= N kλ +1 N kλ +1 > a kλ N kλ >= N kλ N kλ 1 > [a kλ,a k' λ ' ] = δ kk',λλ' For the molecular degrees of freedom, we just pick the molecular eigefuctios r >= ϕ mol > with eergy E. I the ucoupled case, the total wavefuctio is the just a product wavefuctio Ψ tot = ϕ mol (r ) > N kλ > 10-8 Whe the molecule ad eld are iteractig, the real wavefuctio will of course be a superpositio of several such states with differet E ad N kλ values, util the iteractio is over. The overall eergy of molecule ad eld must however always be coserved. 10. Trasitio momets for absorptio, stimulated emissio, ad spotaeous emissio We ow use the Hamiltoia i eq ad the basis i eq to calculate trasitio matrix elemets, the squares of which yield spectral itesities. First, we review three approximatios which we will make here: i) Dipole approximatio: The molecular dimesio r << λ for small molecules. I that case the expoetials i eq ca be approximated as k r <<1 e ik r 1. This is tatamout to sayig that all parts of the molecule experiece the same phase of the icomig electromagetic radiatio. It will be see that this yields the well-kow dipole iteractio µ. E. For electric quadrupole or magetic dipole iteractios, which are rarely domiat i molecules (ad will therefore ot be cosidered here), oe has to let at least e ik r 1 + ik r. For X-rays scatterig, or optical scatterig o macromolecules of size comparable to the wavelegth, oe has to use the full scatterig phase e ik r, discussed i more detail i sectio 1. ii) Neglectig H () H () yields -proto processes i lowest order, e.g. aa results i -photo absorptio, aa i scatterig or Rama processes. H () is ot eeded to describe 1-photo processes such as absorptio or spotaeous emissio. We will later derive a Rama scatterig formula from H (1), k,λ 10-

3 but it should be kept i mid that this formula must be corrected by addig the cotributio from H () i order to yield quatitative results. iii) Golde Rule For a system decayig from a sigle level to a dese maifold of states, the decay rate is give to rst order by Fermi's Golde Rule Γ = π ρ tot < Ψ f H (1) Ψ i > ; ρ tot ρ eld 10-9 as derived from rst-order time-depedet perturbatio theory. Moreover, except for very small cavities, the desity of eld modes is much higher tha the desity of molecular modes, ad we ca approximate ρ tot by just the desity of eld states. The Golde Rule is of course ot exact at log times. A treatmet without the Golde rule has bee give by S. Bergma, J. Math. Phys. 8, 159 (1967) based o the Wiger-Weisskopf method. However, applicatio of the Golde Rule causes oe to miss oly esoteric pheomea at very log fluorescece lifetimes. Takig the expectatio value of the secod lie of eq with the wavefuctio i eq yields < Ψ f H (1) Ψ i >= q - { hc 10-10,k,λ m c Vk }1/ < ϕ f p > ê kλ {< N ' kλ â kλ N kλ > + < N ' kλ a + kλ N kλ >} which is the probability of makig a trasitio betwee two molecular eigestates by chagig the umber of photos i a sigle eld mode (whose eergy must equal the molecular eergy differece i order to assure eergy coservatio). This is the kid of trasitio probability of iterest if a high-resolutio laser is scaed across a molecular trasitio. We ca brig this ito a somewhat more stadard form by usig the idetity p [r, H mol ] = [r, ] = i p ˆ m m Replacig p i eq by the commutator q q < ϕ f p >= m m c m c i (< ϕ f r H mol > < ϕ f H mol r >) = i c < ϕ f µ mol > (E f E i ) 10-1 sice q r = µ mol Isertig eq ito eq.10-10, < Ψ f H (1) Ψ i >= iω k,λ c { hc Vk }1/ < ϕ f µ mol > ê kλ (< N ' a kλ N > + < N ' a + kλ N >) I this expressio, ω is the eergy differece betwee the al ad iitial molecular states (eq. 10-1) expressed i frequecy uits. So ω mol = ω ca always be take as positive, a (-) sig is itroduced i the a term istead of lettig ω em = -ω abs. 10-3

4 We previously determied the classical eld operator, E kλ i terms of vector potetials a ad a * (sectio b), ad ca ow write it i terms of the quatized raisig ad lowerig operators a ad a +. From eq. -7 follows Ê = iω c {hc Vk }1/ a kλ ê kλ + iω c {hc Vk }1/ a + kλ ê kλ = Êkλ + Êkλ = iω abs { hc c Vk }1/ a kλ ê kλ iω em c {hc Vk }1/ a + kλ ê kλ Clearly the eld operator is aalogous to the ordiary harmoic oscillator mometum operator i(a a). The operator a kλ allows for absorptio of a photo at frequecy ω abs = ω eld = kc of + a give polarizatio, a kλ allows for emissio. Isertig eq ito 10-14, we obtai the al formula for the matrix elemet < Ψ f H (1) Ψ i >= < ϕ f µ mol ϕ i > < N' kλ E ˆ N kλ > k, λ which is directly aalogous to the classical expressio from sectio 8 for the iteractio eergy, = µ E kλ k,λ 10.3 Molecular emissio ito a isotropic radiatio eld I this example applicatio of eq , we will use the Golde Rule, assumig the eld desity domiates: Γ f i = π ρ al < Ψ f H(1) Ψ i > The molecular desity of states makes a egligible cotributio to ρ al, which is due almost solely to photo states i the box. ρ al as the volume of the box icreases, but this should be exactly caceled by the V -1/ depedece of H (1) i eq To compute ρ eld, let us assume the eld lls a cubic box with sides of legth L (the al result should of course be idepedet of the shape ad size of the box for large boxes). I that case k i (i=x,y,z) takes o a discrete set of values k i = iπ L, i = 1,, Accordig to g. 10-1, the umber of eld states i a iterval dk i solid agle dω is dn = k dkdω ( π L )3 = V π 3 k dkdω 10-0 To obtai the desity of modes i a give solid agle dω, we itegrate eq over k while eforcig overall eergy coservatio by requirig that ω = E f -E i = ck. This is strictly valid 10-4

5 k z k dk Fig Two-dimesioal cut through the lattice of k-vectors; the icremetal umber of states dn is the umber of lattice poits i the shell dk at k. k x π/l oly if the excited state has a iite lifetime (ad hece a iitely sharp trasitio), but is a good approximatio if the liewidth is sufcietly arrow so that the desity of states does ot vary much over the liewidth: dρ dω = k dk V δ(e ck) 3 π = k V (states per srad per uit E) π 3 c The secod igrediet i eq is the couplig matrix elemet <f H (1) i>: <f H (1) i>= µ ê kλ {N kλ + 1} 1/ ( iω c ){π c Vk }1/ for oe specic k, λ. 10- The rate of emissio ito a agle dω at resoace is thus: Γ f i = π k V π 3 c (N ω π c k, λ c Vk dω µ ˆ e kλ = 4ω3 (N k,λ µ πc 3 e ˆ kλ dω 10-3 To obtai the isotropic emissio rate, we sum over both polarizatios ad itegrate over all agles dω: 4ω Γ tot = 3 (N dω k,λ µ λ =1, πc 3 e ˆ kλ = 4ω 3 π / π / ( N k,λ siθdθdϕ ( πc µ 3 e ˆ kλ )( e ˆ kλ µ ) = 4ω3 (N k,λ + 1) πc 3 = 4ω 3 ( N k,λ πc 3 = 4ω3 (N k,λ + 1) πc 3 = 4ω 3 ( N k,λ πc siθdθdϕµ ( ˆ λ λ e kλ ˆ e kλ )µ si θdθdϕµ (I k ˆ k ˆ ) µ siθdθdϕ{ µ (µ k ˆ ) } si θdθdϕ{ µ µ cos θ} 10-4 = 4ω 3 ( N k,λ { π πc 3 µ π 6 µ } The step i lie 10-4 is possible because e ˆ k 1, e ˆ k, ad k form mutually orthogoal complete set spaig three-dimesioal space; k ˆ is a uit vector poitig i the same directio as k. Note that all k ca be recovered for positive values of the i (Fig. 10-1) so the dω itegratio is oly over the rst octat of the sphere. For simplicity we take the dipole alog the z-axis, but ay 10-5

6 oter axis yields the same result sice the eld uiformly lls the box. We ow have a expressio for the isotropic emissio rate of a molecule at short times, Γ tot = 4ω3 3 c (N µ 3 k,λ Eve whe N k,λ =0, emissio still takes place: this emissio i the absece of a stimulatig radiatio eld is kow as spotaeous emissio. Oe ca thik of it as beig iduced by the zero-poit eergy i the radiatio eld. The spotaeous lifetime is simply the iverse of 10-5 whe N=0, or 3 c 3 τ sp = 4ω 3 µ Excited state lieshape If eq represets the emissio rate i uits of s -1 from a excited molecular state f>, the populatio of molecules i that state f> will decrease as d dt = Γ or (t) = 0 e Γ t 10-7 Amplitude thus leaks away eve though f> is a molecular eigestate because f> is ot a eigestate of the molecule-eld Hamiltoia. I terms of the amplitude of state f>, we ca rewrite eq as f (t) >= e iω t Γ t f (0) > The Γ/ arises because the populatio is give by <f(t) f(t)>, which elimiates the time evolutio phase factor ad restores a exp[-γt] decay for. Fourier trasformig eq yields the lieshape of the trasitio i the frequecy domai, or g(ω) = 1 Γ / π (Γ / ) + (ω ω ) 10-9 istead of δ(ω ω ) / = δ (E E ), which was the lieshape iitially assumed i sectio c) to obtai the desity of states. If the lieshape is sufcietly arrow, the desity of states will ot vary across it as a fuctio of eergy, ad the symmetrical Loretzia 10-9 is a very good approximatio. For very broad lies, the derivatio i c) ad d) is ot self-cosistet, ad the variatio i the desity of states would have to be take ito accout (e.g. Bergma referece i b-iii) Absorptio of a moochromatic polarized plae wave by radomly orieted molecules I high-resolutio spectroscopy, a laser beam is well approximated by a plae wave with a sigle frequecy ω, wavevector k ad polarizatio λ. We derive the absorptio rate i the same two steps as doe for emissio i c), by calculatig the desity of eld modes ad the trasitio matrix elemet. ρ = δ(e ω) / 4π is the ideal case of a truly moochromatic laser: all photos are i the same state so ρde=1, but the desity of states is iite i the iitesimal eergy iterval ear ω. A real laser might have a ite badwidth represeted by a Gaussia or similar lieshape, but we will cosider the ideal case rst. The matrix elemet accordig to eq is <f H (1) i>= (µ ê kλ )N 1/ kλ ( iω c )(π c ad the Golde Rule rate therefore becomes Vk )1/

7 Γ obs = π δ(e ω) ω π c N kλ 4π c Vk (µ ê kλ ) Assumig the molecules are radomly orieted, we average over orietatios by itegratig over 4π steradias. Note that ulike eq. 10-4, where we averaged over all wavevector orietatios i the positive octat, here we average over all possible dipole orietatios. Pickig the laser polarizatio e ˆ kλ alog the z-axis Γ abs = π δ(e ω) ω π π π c N kλ siθ dθ dϕ µ 4π c cos θ Vk 0 = π δ(e ω) ω π c 4π δ(e ω)n kλ 4π c Vk 3 µ = 4π 10-3 ω 3V N kλδ(e ω) µ This ca be rewritte i terms of the laser itesity. For a travelig electromagetic wave, eergy I( area time ) = # photos photo eergy beam velocity = N ω kλ c beam volume V from which follows < Γ abs >= 4π 3 c I µ δ (E ω), the absorptio rate per molecule per secod. I reality, the laser will always have a ite liewidth. So will the molecule, at the very least due to spotaeous emissio. As see i d), the delta-fuctio lieshape the becomes the Fourier trasform of a expoetial decay i the simplest case, or a Loretzia: 1 δ (ω ω ) g(ω ) = 1 Γ / Fi π (Γ / ) + (ω ω ) The 1/ factor is icluded sice δ(e ω ) = δ(ω ω) / ; the 1/π factor is a ormalizatio factor sice τ 1 dδω = π, but τ + Δω δ ( ω ω)dω = The Loretzia thus has the same itegrated liestregth as the delta-fuctio lieshape, but spreads it over a ite frequecy iterval. If Δω laser << τ 1 sp, ad the molecular eergy levels oly decay by spotaeous emissio, the lieshape will be completely domiated by the molecular lifetime: < Γ abs > ( 1 sec ) = πc 1 I τ sp (τ ω 3 τ sp τ sp / 4 + (ω ω ) laser >> τ sp ) Note that the absorptio ad spotaeous emissio rates (iverse of eq. 10-6) are ot idepedet of oe aother. This is required by detailed balace ad eergy coservatio, ad a special case of the Eistei A/B coefciets discussed further i sectio Absorptio coefciet Whe light propagates through a absorbig medium, the itesity geerally drops off expoetially with distace. This is due to the fact that i the absece of saturatio effects, absorptio over a small pathlegth is proportioal to the umber of molecules i the path, ad hece proportioal to the pathlegth Δz:

8 di = Iαdz I = I 0e αz α = di Idz ( 1 cm ) The chage i itesity is give i terms of the absorptio rate by di( eergy cm eergy ) = c( ) du( ) cm sec sec cm 3 = c < Γ abs > ( sec) 1 0 ( molecules )dt ω cm 3 =< Γ abs > 0 dz ω (cdt = dz) Isertig ito ad the result ito yields the absorptio coefciet α(ω) = < Γ > ω abs 0 = 4π µ ω 0 g(ω) I 3c i uits of iverse legth Calculatig the µ It remais to calculate the dipole momet expectatio values µ from the molecular wavefuctio. For electro coordiates r e ad uclear coordiates r, the total dipole operator is give by µ = r e + Z r e I the B.-O. approximatio, Ψ tot > Ψ e > vj >, where the secod ket refers to the vibratiorotatio wavefuctio. Takig the matrix elemet of 10-41, < Ψ f µ Ψ i >=< v f J f < Ψ ef µ Ψ ei > v i J i > 10-4 = < v f J f µ (Q ) v i J i > The electroic expectatio value i eq gives us µ (Q ), the trasitio dipole operator betwee two electroic states as a fuctio of uclear coordiates. µ = µ ii (Q ) is the permaet molecular dipole momet if the electroic states are oe ad the same. If Ψ ei Ψ ef, we have a electroic trasitio. If Ψ ei = Ψ ef, the result is a rotatio-vibratio (v f v i ) trasitio or a rotatioal (v f = v i, J f J i ) trasitio or a e structure/hypere (v f = v i, J f =J i ) trasitio. I all cases, the trasitio dipole operator ca be expaded i a power series i the ormal coordiates Q : µ (Q ) = {µ (0) + Q µ + }ˆ µ (θ,ϕ,χ) Q Q = 0 where µ ˆ is a uit vector poitig i the same directio as µ (Q ) to separate the rotatioal part of the dipole momet operator from its magitude. For electroic trasitios, the leadig term i eq is sufciet to yield a ozero result for the trasitio itesity. Its matrix elemet is µ < v f J f µ (0) µ ˆ v i J i >= µ (0) < v f v i >< J f µ ˆ J i > The rst factor is the electroic trasitio dipole momet at the equilibrium geometry. The secod is the Frack-Codo factor, which requires the vibratioal wavefuctios i two differet electroic states to have ozero overlap for a trasitio to occur. The third factor is the rotatioal expectatio value. Which compoets of the vector µ ˆ actually eed to be icluded depeds o the polarizatio of the electric eld. For example, if E is polarized i the space-xed z-directio, the µ ˆ Z = µ x x ˆ Z ˆ + µ y y ˆ Z ˆ + µ zˆ z Z ˆ is the oly required compoet i 10-8

9 eq , where µ x + µ y + µ z =1. If the trasitio dipole poits alog the molecular z axis (e.g. diatomic molecule or parallel symmetric top bad), the oly ecessary compoet of µ ˆ i eq is z ˆ Z ˆ =cosθ. I sectio 5 it was see how to evaluate matrix elemets of such operators. Withi a give electroic surface, eq becomes µ(q ) = {µ(0) + Q µ + }ˆ µ (θ,ϕ, χ) Q Q =0 where µ is the permaet dipole momet operator o that electroic surface. For pure rotatios, the rst term is agai sufciet to give the leadig cotributio to the lie itesity: < vj f µ vj i > < v µ(0) v >< J f µ ˆ J i >= µ(0) < J f µ ˆ J i > µ(0) is the permaet dipole momet, ad the rotatioal matrix elemet is evaluated as discussed above. For ro-vibratioal trasitios, the leadig term i makes o cotributio, because vibratioal wavefuctios withi a give electroic state form a orthogoal set, ad µ(0) is just a costat. Therefore, µ < v f J f µ v i J i > < v f Q v i >< J f µ ˆ J i > Q The derivative times the vibratioal matrix elemet i is the vibratioal trasitio momet. Note that ear the bottom of the well, the vibratioal eigefuctios are early harmoic oscillator fuctios; i that case the vibratioal matrix elemet yields a selectio rule of Δv = ± 1, e.g.: < v f J f µ v i J i > µ { } 1/ (v i 1/ δ Q ω < J v f = v i +1 f µ ˆ J i > Higher vibratioal eigefuctios are ot harmoic-oscillator-like due to aharmoicity, ad so overtoe trasitios become allowed. (Q terms would have to be icluded i the expasio i eq to give quatitative overtoe itesities.) 10.8 Higher-order processes We close the discussio o secod quatizatio by briefly cosiderig some higher order processes. There are two types: i) Multipole trasitios: e ik r =1 + ik r+... the secod term leads to a matrix elemet i < ϕ f p k r ϕ i > r ω if the same commutator trick as i eq is applied. Based o sectio 9, these terms will deped o the quadrupole momet or o the magetic dipole momet. Such trasitios are commo i atoms. Molecules have lower symmetry, ad are more likely to have dipole trasitios. There are cases where dipole-forbidde trasitios have bee observed. However, their itesity is usually domiated by vibroic (o-bor-oppeheimer) iteractios with a dipole-allowed state. Thus, magetic dipole/electric quadrupole trasitios are almost ever a factor i molecular lie itesities, uless the molecule is a macromolecule. ii) Multiphoto trasitios: by doig secod order perturbatio theory o the iteractio Hamiltoia 10-14, the ext higher correctios to the trasitio momet are obtaied ad the Golde Rule rate becomes Γ f i π ρ < Ψ H (1) f Ψ i > + < Ψ f H () Ψ i > + j i, f <Ψ f H (1) Ψ j ><Ψ j H (1) Ψ i > E i t E j t The rst term has already bee cosidered extesively. The secod two terms ivolve a product + of two a kλ / a kλ ad are therefore two photo trasitios because they chage two quata of the 10-9

10 eld. I processes that are ot oe-photo resoat, the rst term is usually egligible, ad oly the last two eed be cosidered. Furthermore, the secod term is ot importat i cases where trasitios are ehaced by ear-resoace, sice H () cotais o molecular coordiates or mometa. We will therefore cosider the third term as a example. I a spotaeous Rama process a laser excites the molecules iitially i state ϕ i > at frequecy ω, ad light is emitted at a frequecy ω', returig the molecule to differet state ϕ f >. The itermediate molecule state ϕ j > eed ot be resoat with the iput photo as diagrammed i g A photo at frequecy ω could be absorbed "virtually" although o state is resoat, ad the a secod photo ω' could be emitted to a molecular state at higher eergy tha the origial state (Stokes trasitio). j> ω ω' virtual f> Fig Diagram for the two-photo spotaeous Rama process; the state E j is oresoat. i> The two types of itermediate processes i such a case ca be diagrammed as (tot E ) j =E j ω >, N kλ,0, > ϕ j >, N kλ 1,0, > ϕ j >, N kλ,1, > E j (tot ) =E j + ω ' ϕ f >, N kλ 1,1, > I the upper path, photo ω is absorbed before photo ω is emitted, ad the coverse occurs i the lower path. Note that the umber of modes i the box cotaiig the molecule is typically so great, that N kλ 0 for ay give mode if oly black body radiatio is preset. I the spotaeous Rama process, the itermediate state is usually a excited electroic state, so µ ji µ ji (0) < v j v i >< J j ˆµ ji J i >. Near resoace, the third term i eq is largest ad yields Γ f i = π ρ e ˆ k' λ' µ fj µ ji e ˆ k λ Ei (E j ω ) iω ji c {hc Vk }1/ e ˆ k' λ' µ fj µ ji e ˆ k λ ( iω ji E i (E j + ω' ) c ){hc Vk' }1/ N kλ ( iω fj c ){ hc Vk' }1/ 1( iω fj c ){hc Vk }1/ N k λ If the iput laser is polarized, the output itesity may ot be isotropic due to aligmet of the excited molecular dipoles. Otherwise, the total desity of states is approximately the product of the laser desity of states at frequecy ω ad the desity of states available i the box (black body cavity) where the experimet is doe: ρ ρ laser (ω)ρ box (ω ') δ(ω ω) laser k ' V 4π π 3 c (uits: Joule 1 Hz 1 )

11 Note that oe of the desities of states is expressed as states per frequecy iterval istead of states per eergy iterval (factor of differece) for the ext step. Isertig eq ito eq Γ (s 1 Hz 1 ) = π δ(ω laser ω) k ' V h ωω ' N kλ ê 4π π 3 c V k ' λ ' R ê kλ, 10-54a ω '3 c ω N = kλ δ(ω πc 4 laser ck) V ê k ' λ ' R ê kλ, 10-54b ω '3 = π c I(ω)( eergy 4 areaitimeihz ) ê k ' λ ' R ê kλ (cm 6 ) 10-54c where R is the polarizability tesor R (cm 3 ) = { µ µ fj ji + µ µ fj ji E ji ω E ji + ω ' }ω fjω ji ω ω ' j ad I(ω) is the iput laser itesity per uit badwidth of the laser (i.e. the eergy per uit area ad uit time, per 1 Hz of laser badwidth). Expressig the laser itesity per frequecy iterval istead of the total laser itesity is particularly useful if the laser has a ite badwidth, i.e. it is ot the ideal delta fuctio electric eld i 10-54b. Note that Γ is the rate per uit frequecy badwidth of the pump laser, ad s -1 Hz -1 is uitless. Note that the prefactor i eq c ca differ slightly if the experimetal geometry is chaged. However, absolute Rama trasitio rates are ever measured i practice, ad the importat factors are that the sigal depeds o the cube of the scatterig frequecy (favorig a blue rather tha a red pump laser sice ω ' = ω + E ji ), that the sigal depeds liearly o the pump laser itesity, ad o the square of the polarizability tesor R. The itesity term i eq c could be split ito two electric eld cotributios yieldig H () ~ (Ee ˆ R) E = µ id E Oe ca therefore thik of the spotaeous Rama effect as beig due to a off-resoat eld at frequecy ω iducig a dipole momet i the molecule, which the iteracts as usual with the eld