1. Chapter 7. Emission and Absorption and Rate Equations
7.1 Introduction
For most considerations b (total relaxation rate)
is much faster than the rate at which external
forces cause electron to jump between atomic
energy levels.
The result of the external force,
F=-eE is only to produce a gradual
increase or decrease in probability.
(6.5.18)
(6.5.19)
Therefore, such a fast phenomena can often be
treated with sufficient accuracy in an average sense.
Absorption rate / Stimulated emission rate
(Rates of increase and decrease in probability)

2. 7.2 Stimulated Absorption and Emission Rates
In Chapter 6, density matrix equations considering the relaxation effects are given by
(6.5.14)
(6.5.17)
When
(condition for adiabatic following to occur),
quasisteady-state approximation,
is possible ;
(6.5.17) =>
(7.2.1)
: Adiabatic solution ※
adiabatically follow the
inversion

3. <Stimulated Absorption and Emission Rates>
absorption or emission rates
(6.5.14) =>
(7.2.2)
: Population rate equation ※
are coupled only to
each other
<Stimulated Absorption and Emission Rates>
1) For nondegenerated transitions,
(7.2.3)

4. Induced transition rates (abbreviation), :
Calculation of |c|2
In many cases (unpolarized radiation, rotational or collisional disorientation, etc),
orientational average of |c|2 is simpler and useful
(7.2.4)
Homework : Problem 7.1
where,
: Complex dipole moment and its projection on
In terms of cartesian components,
Induced transition rates (abbreviation), :
(orientation-averaged)
(7.2.7)

5. Absorption cross section, :
[Refer to (7.4.2)]
(7.2.8)
where,
: photon flux
2) For degenerated transitions (Homework : Refer to Appen. 7.A)
: In the case of natural
excitation (the # of atoms
in each of the different
degenerated states of the
same level are equal) ; 2
g2=5 1
g1=3
<Example of degenerated transition>

6. 7.3 Population Rate Equations
Densities of atoms in levels 1 and 2 ;
(7.3.1)
where,
: total density of atoms
(7.2.2) =>
(7.3.2) ※
: This indicates that ineleastic collisions will takes all of the atoms
out of levels 1 and 2 into other atomic levels. Nevertheless,
are practically small relative to , and the intermediate time
behavior is of the most interest.
=> We can ignore the

7. (7.3.4)
(No inelastic collision), then
Sol)
(7.3.7)

8. Examples) 1) No radiation field ; , 2) Weak radiation field ; ,※
: Lorentz classical theory is valid.
3) Strong radiation field ; ,

9. [Fig. 7.1]

10. Power Broadening
In the limit,
(7.3.7) =>
Half width ;

11. 7.4 Absorption Cross Section and the Einstein B Coefficient
(7.2.8),
(7.2.7) =>
Put,
where,
: Lorentzian line shape function
: generaliztion for arbitrary line shape function

12. Examples) 1) For descrete radiation frequencies
For a single frequency, :
2) For continuous band radiation
(narrow band limit) If
(broad band limit)
: Einstein’s
empirical definition

13. 7.5 Strong Fields and Saturation
What is the criterion for “strong” field ?
=> Saturation the population ;
This criterion is satisfied in (7.3.13), if
Define,
ex)

14. 7.6 Spontaneous Emission and the Einstein’s A Coefficient
An atom in an excited state will eventually drop to a state of lower energy,
even in the absence of any field or other atoms. => Spontaneous emission (※
Spontaneous emission would occur even for a single excited atom in a perfect vacuum !)
ex) Luminescence, Fluorescence, Phosphorescence
(7.3.8) =>
: characteristic time constant (excited state “life time’)
- In the case that there are multi-channels for radiative transition,

15. Quantum mechanical expression ;
(7.6.4)
(7.6.5)
Homework : Problem 7.3
where,
Line shape for the spontaneous emission : Lorentzian
(7.6.7)
where,
: Natural linewidth