Download Spectra of Diatomic Molecules: a Summary

Spectra of Diatomic
Molecules, a Summary
Molecular Physics, Wolfgang Demtroder
Questions regarding this topic are
in red below….
1. Between which pairs of molecular states
can transitions take place by absorption
or emission of electromagnetic radiation?
2. What is the transition probability and
what are the factors by which it is
determined?
3. What are the spectral profiles of emission
or absorption lines for such a transition?
Einstein Coefficients
Absorption: The Probability
Stimulated Emission
Spontaneous Emission
𝑑𝑊𝑖𝑘
𝑑𝑡
𝑑𝑊𝑘𝑖
𝑑𝑡
𝑑𝑊𝑘𝑖
𝑑𝑡
= 𝐵𝑖𝑘 𝜌(𝑣
𝑎𝑏𝑠
𝐵𝑖𝑘
= 𝐵𝑘𝑖 𝜌(𝑣
𝑠𝑡𝑖𝑚
= 𝐴𝑘𝑖
𝐴𝑖𝑘
𝑔𝑘
=
𝐵
𝑔𝑖 𝑘𝑖
8𝜋ℎ𝜈 3
=
𝐵𝑖𝑘
𝑐3
𝑠𝑝𝑜𝑛
Note:
𝑑𝑊𝑘𝑖
𝑑𝑡
Indicates
probability per unit time
can be larger than one!
𝐴𝑖𝑘 (3𝑝 − 3𝑠) ≈ 108 𝑠 −1
For sodium
Average Power
Dipole Moment
Classical:
Quantum Mechanical:
𝑑 = 𝑞 𝑟 = 𝑑0 si n( 𝜔𝑡)
∗
𝑑 = 𝑒 𝑟 = 𝑒 𝛹𝑖 𝑟 𝛹𝑖 𝑑𝜏
dipole moment of a stationary state
2 𝑑2 𝜔4
𝑃=
3 4𝜋𝜖0 𝑐 3
∗
𝐷𝑖𝑘 = 𝑒 𝛹𝑖 𝑟 𝛹𝑘 𝑑𝜏
𝑃𝑖𝑘
With
𝑑2 =
1 2
𝑑
2 0
4 𝜔𝑖𝑘 4
=
𝐷
3 4𝜋𝜖0 𝑐 3 𝑖𝑘
power emitted
For a transitionAverage
both states
must in the transition
contribute = transition dipole moment
We replace the classical average 𝑑 2 by the quantum mechanical expression
1
𝐷𝑖𝑘 + 𝐷𝑘𝑖 2 = 2 𝐷𝑖𝑘 2
2
• Although the fluorescence intensity of a single molecular
• dipole has an angular distribution I(theta)a sin(theta), where theta is the
angle between the dipole axis and the direction of observation, the total
emission from N molecules with random orientations in space is isotropic.
Electric Dipole Transition for N atoms
atoms radiate
Vibration-Rotation Transitions
Selection Rule:
1
Vibrational Bands
R-band
P-band
During a transition between different vibrational states the
internucleus distance changes and thus also the rotational constant
Difference of
vibrational
levels without
rotation
Electronic Transitions,Franck-Condon Factors
Example
• For R-branch with
• For P-branch with
Frank-Condon Factors
Spectral Line Shapes
Classical Model
Doppler Shift
Gaussian Profile
Voigt Profile
Collisional Broadening
Effects of Elastic and Inelastic Collisions
Two-Photon Transitions
Multi Photon Transitions
Inelastic scattering: Raman Scattering