A number of photons is assigned to every
light wave of the frequency ν.
Further one assumes that there are discrete atomic (or molecular) energy levels.
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hν = E2 - E1 |
There are a lot of transitions occuring equally in both directions (absorption and emission) in the equilibrium position; therefore the population number (?) may not change on average:
dN2/dt = 0 (sum of the three processes)
dN2/dt = B12 u(ν)N1 − B21 u(ν)N2 − A21N2 = 0
® B12 u(ν)N1 = (B21u(ν)+A21)N2
(B12u(ν))/(A21+B21u(ν)) | = | N2/N1 | = | e−E2/kT/e−E1/kT | = | e−hν/kT |
→ | thermal equilibrium, thus Boltzmann distribution |
u(ν) = A21/(B12 ehν/kT − B21)
The coefficients A12, B12 and B21 are determined from experiments:
Þ | B12 = B21 | → | u(ν) = (A21/B12)/(ehν/kT− 1) |
A21/B12 = 8πhn ³/c³ |
The ν3 relation between spontaneous
emission and absorption is very important here !
If we summarize all our knowledge about the coefficients B12,
B21 and A21 ,we will obtain the Planck
Radiation formula:
u(ν) = 8πhν³/c³.1/ehν/kT− 1 |
So, by the way we have also approached an
understanding of operation principle of Lasers;
since we must only interpret the above formula {dN2/dt
= B12 u(ν)N1 −
B21 u(ν)N2 −
A21N2} in another way:
Therefore we have lasing whether the photon production is alltogether more
than 0. So if u(ν) increases, N2
should decrease, i.e. dN2/dt
< 0. This is important for the laser operation:
or, if we neglect the spontaneous emission there should be always the following
N2 > N1,
i.e. the population number (?) of the upper state should be more than the population number of the lower state. However it's not possible in the thermal equilibrium position as one can see from the Boltzmann energy distribution. Thus lasing can be reached only in the thermodynamically non-equilibrium position.