Einstein Rate Equation Model
Radiation in a two-level system
E2, n2, g2
E2  E1  h
Cu
A
Bu
E1, n1, g1
u
the incident radiation field (mode density)
Cu
A
1. Absorption process
Bu
3. Stimulated emission process
2. Emission process
A, B, C are the Einstein coefficients
E2, n2, g2
Rate equation model
Cu
A
Bu
E1, n1, g1
Population of states:
dn2
 Cu n1  An2  Bu n2
dt
Impose an equilibrium condition
dn2
0
dt
Statistical Physics: thermal excitation
nT   exp  E / kT 
Relative population:
n2 exp  E2 / kt
 h 

 exp  
n1 exp  E1 / kt
 kT 
In steady state:
n1 A  Bu

n2
Cu
Atomic two-level system in equilibrium
with radiation field:
u 
A
C exp h / kT   B
Results from the Einstein model
E2, n2, g2
Cu
A
Bu
E1, n1, g1
A two-level atom in equilibrium
with a radiation field:
The B constant related to the
Transition dipole moment
(the QM radiation strength)
CB
The A-constant follows from
Stimulated emission is equally strong
as absorption
A 8h 3

B
c3
Properties of a two-level system
Under all circumstances, i.e.
for arbitrary radiation fields
u
Cu  A  Bu
Emission is stronger than absorption
Optical pumping can never result
in a population inversion
So if we start at
n1 (0)  N
(all population in the ground state)
It is not possible to reach:
n2  n1
A two-level LASER is impossible
Decay of an excited state
In the absence of a radiation field:
E2, n2, g2
A
E1, n1, g1
u  0
Rate equation reduces:
dn2
  An2
dt
With a boundary condition;
n2 (0)  N
numbers in 106 s-1
Solution:
n2 (t )  Ne At  Net / 
Note for multiple
decay channels
Lifetime of an excited state

1
A

1
 Ai
i
Quantum states and kinetics
Hydrogen at 20°C.
Estimate the average kinetic energy of whole hydrogen atoms
(not just the electrons) at room temperature, and use the result to
explain why nearly all H atoms are in the ground state at room
temperature, and hence emit no light.
3
K  k BT  6.2  1021 J  0.04eV
2
Gas at elevated temperature – probability exited state is populated:
Pn (T )  e  En / kT
Energy in gas:
E 
 En e
n
e
n
 En / kT
 En / kT