|Wachsmann-Hogiu, Sebastian: Vibronic coupling and ultrafast electron transfer studied by picosecond time-resolved resonance Raman and CARS spectroscopy |
The Raman intensity can be written as:
where is the Raman cross-section for the level , is the population of the level , and is the incident laser intensity.
The measured difference signal normalized to the corresponding Stokes intensity is:
Three cases will be discussed in the following, under the assumption that
The Stokes and anti-Stokes intensities become:
The population of level can be written according to the Boltzmann distribution:
where following notation has been made:
with the Planck constant, the vibrational frequency, the Boltzmann constant, and the temperature.
The sum of the populations over all levels equals the total number of scattering molecules per unit volume :
Inserting (A.7) into (A.5) results:
With (A.8), Eq. (A.3) transforms into:
and Eq. (A.4) becomes:
Here, the equality has been used.
With the notation:
Eq. (A.2) transforms:
In this case, one can write for the stokes and anti-Stokes intensities:
Introducing (A.13) and (A.14) in (A.2), following expression is obtained:
With (A.16) and (A.17), Eq. (A.2) can be written as:
In Eq. (A.18) the notation (A.11) has been used.
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