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10_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.76 Lecture #10 Fall, 2008 Page 1 Lecture #10: Transitions II Last Time oscillating electric field 2 P Intensity if ∝ε 2 v i v f Ω f J f M f M i e r f r b,if Ω i J i M i α Sb θ , φ S,b unique radial universal angular factor all electric factor * polarization: S, M R * band type: b, ∆Ω dipole * branch type J transition probabilities pure rotation i = f b = z for diatomic (µ along z) v i = v f σ (xz) and σ (yz) symmetry b = z ∆Ω = 0 = M z,ii (R e ) + dM 1 d 2 M v M z,ii (R) v R    dQ Q = 0 Q vv + 2 dQ 2 Q = 0 Q vv 2 if homonuclear =0 n Q vv matrix elements in Harmonic Oscillator Basis Set (Q = R – R e ) 2 P if [ const. + small v 2 term ] Ω J f M Ω J i M α Zz µ 2 (dµ/dR) 2 Today: finish pure rotation spectrum Hönl-London Factors rotation-vibration spectrum v = ±1 propensity rule dM/dR 0 anharmonic and centrifugal correction terms [PERTURBATION THEORY] rotation-vibration-electric spectrum all v Franck Condon factors R-centroid approximation stationary phase approximation 2 Ω J f M Ω J i M α Zz Final factor is Each J consists of 2J + 1 degenerate M-components.
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5.76 Lecture #10 Fall, 2008 Page 2 direction cosine matrix elements sum over M Hönl-London rotational linestrength [can't do this sum so simply for OODR because initial M’s are not equally populated] factors see Hougen page 39, Table 7 2 ΩΩ 3 Ω Herzberg Diatomics, page 208 J f M S J i J f Ω J i M α Zz θφ M sum rule 2J f + 1 el 3 g f useful for checking calculations ( J f + Ω + 1 )( J f − Ω + 1 ) J f = ~ 3 J ( f + 1 ) 3 Ω 2 ( 2J f + 1 ) 2 Ω 2 = ~ 3J f 3J f ( J f + Ω )( J f − Ω ) J f = ~ 3J f 3 common final state J i = J f + 1 (R or P) J i = J f ( Q branch weak at high J) J i = J f –1 (P or R) The increase with J is due to 2J + 1 degeneracy factor being included. These formulas for a common final state do not depend on whether J i or J f is upper or lower state. Similar set of formulas for
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