Natural Lineshape
In a semiclassical picture we can look at the emission process in the
following way: When an atom emits a photon, the transiting electron in
the atom oscillates for a finite time at a definite frequency. But because
the oscillation is only for a finite time, it corresponds to a
range of frequencies (Fourier series).
Quantum mechanics and in particular Heisenberg's uncertainty relation allows us a different view of the process. The excited states of atoms
have a limited live time, an atom in an excited state will eventually
spontaneously emit a photon and go into a low lying excited state. The
photon will carry away the energy difference between these two electronic
states. Heisenberg's uncertainty and the limited live time of the excited
state, now implies that the exact energy of these states is not defined.
The photon energy can consequently not be exactly defined either. As the
energy of a photon is linked to the frequency of the associated electromagnetic
field, the frequency of oscillation is not defined with absolute precision.
The frequency spectrum of the emitted electromagnetic wave package has a certain width.
The time of oscillation is typically 10^{8} s for optical
transitions, quite long on an atomic scale when light is travelling at
3x10^{8} m/s. The oscillating at about frequency for a visible
line will be of about 6x10^{14} Hz. The wave package can
therefore extend for the equivalent of 3 m. This length can be associated
to the coherence length. It corresponds to 6 000 000 oscillations.
Hence the range of frequencies is very small and the emitted line is very narrow.
The Lorentzian function and the exponential decay of a spontaneously
decay population of excited atoms are linked to each other by Frourier
transformation. The constant b a parameter for the width of the Lorentzian line is propotional to the inverse of the live time of the excited state.
The natural shape of the line is Lorentzian and it typically
has a width of only 0.01 pm. The lorentzian lineshape is charcterised by it's large wings.
Authors: Richard Payling and Thomas Nelis
First published on the web: 15 November 1999.
