14a - Molecular Spectroscopy General definitions 1 The...

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1 Molecular Spectroscopy General definitions
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2 in I out I / cc ν λν == ± i E f E h Molecular structure A The double-beam spectrophotometer 2 # photons Intensity: cm s I = Initial state Final state 0 0 00 0 0.4343 0.4 0 3 0 43 Transmittan ln log 0.4343 10 =1 0 0 All the light transmitted; All the lig ce: 11 T ht absorbe T=1 = T0 . =0 = d A k b I b b I k dx dx dx b b I dI I dI dI II I kk k k k k k I b b ⎛⎞ =− ⎜⎟ ⎝⎠ ⇒= −⇔ = = ⇒⇒ = =∞⇒ ∫∫ 0 0 -log( ) -log log 0.4343 0 zero T=1 A= T absorption bsorbance: =0 A . complete absorption Extinction coeffic = . ient: k k k I I AT A b b b c ε = = The Beer-Lambert law (absorption spectroscopy) Molar Extinction coefficient Chromophore Concentration (mol/L)
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3 ( ) 00 sin 2 molecule H E H t π µ ν =− ±²²³ ² ²´ Dipole moment operator 0 2 ( ( ) ) f i d ψ τν ε δ ∝− Field-matter interaction (dipole approximation, linear spectroscopy) Electric component of electromagnetic field Field amplitude Field frequency Transition dipole Moment leads to Selection Rules Molecular extinction coefficient Resonance condition Molecular Hamiltonian 2 12 Integration over real space and spin space 1 E 2 E A b s o r p t i n E m Absorption transition rate = Emission transition rate Q: Why do we usually see net absorption? A: Non-radiative processes and fluorescence favor lower lying energy levels, as dictated by the Bolzman distribution at thermal equilibrium 2 E 1 E Non-radiative, Fluorescence 1 1 1 1 2 2 2 22 1 () ( e x p ( exp ) 0 ( ) exp ) eq B q B B e PE E gE kT E E P E E E g ⎛⎞ ⎜⎟ ⎝⎠ = = −> 1 2 ( Abs ) orpt ) : ( ion h E M M E +→ 2 1 Stimulated emission: ( 2 ) ME hh +
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4 Einstein A and B coefficients Absorption Spontaneous emission Stimulated emission 1 E 2 E 1 () N B ρν 2 N B 2 N A [] 2 2 1 2 11 2 1 2 ( a b s o r p t i o n= em i s s i o n ) exp exp (detailed balance) Radiation density: ( ) 1e x p 1 Compare to Planck's Equilibrium law for blackbo y : d BB B N NE N h k B NN E Tk N N A B A hk A T T ν =+ ⎡⎤ == ⎢⎥ ⎣⎦ ⇒= = −− 3 3 3 3 3 81 radiation: ( ) exp 1 ; 8 / rapidly increases with (like ). B h ch k T h c A A πν νν = Line-broadening mechanisms 1 min 44 Nat h h t E E t π ×≥ ⇒ = = (1) The natural line width Life-time of the exited state t (2) The Doppler effect 6 8 0 0 0 1 1 12 1 0 31 0 600 cc m vm s vs = ± = × × Detected frequency Source frequency Source speed + toward detector - away from detector
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5 (3) Collisional broadening ** B B A A + →+ More collisions
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This note was uploaded on 06/13/2009 for the course CHEM 260 taught by Professor Staff during the Spring '08 term at University of Michigan.

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14a - Molecular Spectroscopy General definitions 1 The...

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