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Summary for Week 12

Reading in Levine: Chapter 20

Homework problems

Hand in on Monday, April 20: 20.22, 20.24. 20.26, 20.31, 20.32, 20.36, 20.43, 20.49, 20.50

Key Concepts

Absorption and emission of radiation for a two-state system.

At equilibrium we can write

dN₂/dt = B₁₂N₁ -B₁₂N₂ -A₂₁N₂ = 0

 

By comparing the resultiing expression for with the Planck black body radiation law one obtains:

Dipole selection rules (obtained by evaluation d₁₂ with the relevant wave functions)

Δv = ±1 for harmonic oscillator

ΔJ = ± 1 for rigid rotor

Using the harmonic oscillator and rigid-rotor models to interpret the rovibrational spectral of diatomic molecules.

Homonuclear diatomics have no dipole moment and thus no change in dipole moment and thus no dipole (E1) spectrum

As an aborbed photon carries one unit of angular momentum (h/2π), there is no purely vibrational spectrum for heteronuclear diatomcs. Rather a ro-vibrational spectrum is seen.

ΔE (R-branch) = E(v=1,J+1)-E(v=0,J) = hn₀ + (h²/8π²I)[(J+1)(J+2)-J(J+1)

= hn₀ + (h²/8π²I)2(J+1) for J=0,1,2,3

ΔE (Q-branch) = hn₀ is absent for dipole (E1) radiation

ΔE (P-branch) = E(v=1,J-1)-E(v=0,J)=hn₀ + (h²/8π²I)[(J-1)(J)-J(J+1)

= hn₀ - (h²/8π²I)2J for J=1,2,3,4

P-branch .....................................................R branch

The quantity B=(h/8π²I) is called the rotational constant.where I=µr_e² is the moment of inertia.This means that hB has units of energy (J in SI units)

= B/c =(h/8π²Ic) is in cm^-1 if all quantities are in SI units except c= 3.00x10¹⁰ cm/s

 

 

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