Fano resonance
Encyclopedia
In physics
, a Fano resonance is a type of resonant scattering phenomenon that gives rise to an asymmetric line-shape. Interference between a background and a resonant scattering process produces the asymmetric line-shape. It is named after Italian physicist Ugo Fano
who gave a theoretical explanation for the scattering line-shape of inelastic scattering of electrons off of helium. However Ettore Majorana
was the first to discover this phenomenon. Because it is a general wave
phenomenon, examples can be found across many areas of physics and engineering.
. The incident electron doubly excites the atom to the
state. The doubly excited atom spontaneously decays by ejecting one of the excited electrons. Fano showed that interference between the amplitude to simply scatter the incident electron and the amplitude to scatter via autoionization creates an asymmetric scattering line-shape around the autoionization energy with a line-width very close to the inverse of the autoionization lifetime.
For energies far from the resonant energy the background scattering process dominates. Within
of the resonant energy, the phase of the resonant scattering amplitude changes by 
. It is this rapid variation in phase that creates the asymmetric line-shape.
Fano showed that the total scattering cross-section assumes the following form,
where q, the Fano parameter, measures the ratio of resonant scattering to the direct (background) scattering amplitude. (This is consistent with the interpretation within the Feshbach–Fano partitioning theory.) In the case the direct scattering amplitude vanishes, the q parameter becomes infinite and the Fano formula boils down to the usual Breit–Wigner (Lorentzian) formula:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a Fano resonance is a type of resonant scattering phenomenon that gives rise to an asymmetric line-shape. Interference between a background and a resonant scattering process produces the asymmetric line-shape. It is named after Italian physicist Ugo Fano
Ugo Fano
Ugo Fano was an Italian American physicist, a leader in theoretical physics in the 20th century.- Biography :Ugo Fano was born into a wealthy Jewish family in Turin, Italy...
who gave a theoretical explanation for the scattering line-shape of inelastic scattering of electrons off of helium. However Ettore Majorana
Ettore Majorana
Ettore Majorana was an Italian theoretical physicist who began work on neutrino masses. He disappeared suddenly in mysterious circumstances. He is noted for the eponymous Majorana equation and for Majorana fermions.-Gifted in mathematics:Majorana was born in Catania, Sicily...
was the first to discover this phenomenon. Because it is a general wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
phenomenon, examples can be found across many areas of physics and engineering.
History
The explanation of the Fano line-shape first appeared in the context of inelastic electron scattering by helium and autoionizationAutoionization
Autoionization is a process by which atoms or molecules spontaneously emit one of the shell electrons, thus going from a state with charge Z to a state with charge Z + 1, for example from an electrically neutral state to a singly ionized state....
. The incident electron doubly excites the atom to the
state. The doubly excited atom spontaneously decays by ejecting one of the excited electrons. Fano showed that interference between the amplitude to simply scatter the incident electron and the amplitude to scatter via autoionization creates an asymmetric scattering line-shape around the autoionization energy with a line-width very close to the inverse of the autoionization lifetime.Explanation
The Fano resonance line-shape is due to interference between two scattering amplitudes, one due to scattering within a continuum of states (the background process) and the second due to a excitation of a discrete state (the resonant process). The energy of the resonant state must lie in the energy range of the continuum (background) states for the effect to occur. Near the resonant energy, the background scattering amplitude typical varies slowly with energy while the resonant scattering amplitude changes both in magnitude and phase quickly. It is this variation that creates the asymmetric profile.For energies far from the resonant energy the background scattering process dominates. Within
of the resonant energy, the phase of the resonant scattering amplitude changes by . It is this rapid variation in phase that creates the asymmetric line-shape.Fano showed that the total scattering cross-section assumes the following form,
where q, the Fano parameter, measures the ratio of resonant scattering to the direct (background) scattering amplitude. (This is consistent with the interpretation within the Feshbach–Fano partitioning theory.) In the case the direct scattering amplitude vanishes, the q parameter becomes infinite and the Fano formula boils down to the usual Breit–Wigner (Lorentzian) formula: