Comparison With an Experiment

We have learnt such theoretical quantities as transition moment µmn, matrix element Rmn, oscillator force fmn and Einstein factors Amn and Bmn for spontaneous and induced emission and also Bnm for absorption in the chapter "Optical Transitions". Now we want to obtain these quantities in an experiment. First, we will consider the time-resolved emission measurement and then "common" absorption measurement.

The excited particle change in time is given by:

dNm/dt = −Amn Nm
→ Nm(t) = Nm(t=0) e−A mnt

Nm(t=0) is the amount of particles (per volume) for zero time t = 0 and dNm/dt is proportional to the amount of emitted photons pro time. The lifetime [s] of a state is defined by Amnt = 1:

t = 1/Amn

The transition probability Amn is directly inverse proportional to the state lifetime when nothing happens with this state (for instance, collisions with other particles, internal conversion and so on). Nevertheless, each state perturbation decreases the lifetime giving higher transition probability. If we determine Amn we can obtain other theoretical quantities from it.

After Beer-Lamberts law we can write down for the transmitted intensity Itrans of incident light Io coming through some density of particles N:

Itrans = Io e-s N l

σ: absorption coefficient/cm²
N: number of particles/cm³
l: absorption length/cm

We can determine the absorption coefficients σ from experiment:

σ = 1/N l . ln( Io/Itrans)

σ also depends on frequency ν of the transition. The total intensity of the transition that is òsdν can be expressed through Bnm:

σo = òsdν = Bnm · hνmn/c

Here we have connected the experimentally observed quantity òsdν which is also referred to as the integral absorption cross section σo and theoretical quantity Bnm.

Chemists like to measure light-absorption quantity ε[Liter mol-1cm-1] using the following formula

Itrans = Io · 10-ec l

where c is concentration [mol/Liter]. One can easily obtain relation between ε and σ comparing with the Beer-Lamberts law:

ln Itrans/Io = - e c l ln10 = - σ N l

Since N is in molecule/cm³ and c is in mol/l we obtain the following after some recalculations (1 Liter = 1000 cm³):

ε = σ · N/c 1/ln10 = σ · NA/1000·ln10

Where NA is the Avogadro constant (number of particles pro mol)

ε [dm³ mol-1cm-1] ≈ 2,62 1020 σ [cm² Molecule-1]

Here one can find recalculations between some experimental quantities.

Here one can find summary of all formulas:

σo = B hν/c = πe²ν/3εo hc |R|² = e²/4εo cm e · f
B = c/hν · σo = e²/6εo h² |R|² = e²/4εo hνm e · f
|R|² = 3εo hc/πe²ν · σo = 6εoh²/e² B = 3h/4pnme · f
f = 4εo cm e/e² · σo = 4εo hνm e/e² · B = 4pnme/3h |R|²

It is all integral quantities for spectral transition with middle frequency ν. How does the s(n)-dependence look like, however? Here one can find the answer.