Shape resonance
Encyclopedia
A shape resonance is a metastable state in which an electron is trapped due the shape of a potential barrier.
Serhan describes a state as being a shape resonance if, "the internal state of the system remains unchanged upon disintegration of the quasi-bound level."
A more general discussion of resonances and their taxonomies in molecular system can be found in the review article by Schulz,; for the discovery of the Fano resonance
line-shape and for the Majorana
pioneering work in this field by Antonio Bianconi; and for
a mathematical review by Combes et al..
, a shape resonance, in contrast to a Feshbach resonance
, is a resonance
which is not turned into a bound state
if the coupling between some degrees of freedom
and the degrees of freedom associated to the fragmentation (reaction coordinate
s) are set to zero. More simply, the shape resonance total energy is more than the separated fragment energy.
Practical implications of this difference for lifetimes and spectral widths are mentioned in works such as Zobel.
Related terms include a special kind of shape resonance, the core-excited shape resonance
, and trap-induced shape resonance.
Of course in one-dimensional systems, resonances are shape resonances. In a system with more than one degree of freedom, this definition makes sense only if the separable model, which supposes the two groups of degrees of freedom uncoupled, is a meaningful approximation. When the coupling becomes large, the situation is much less clear.
In the case of atomic and molecular electronic structure problems, it is well known that the self-consistent field
(SCF) approximation is relevant at least as a starting point of more elaborate methods. The Slater determinant
s built from SCF orbitals (atomic
or molecular orbital
s) are shape resonances if only one electronic transition is required to emit one electron
.
Today, there is some debate about the definition and even existence of the shape resonance in some systems observed with molecular spectroscopy. It has been experimentally observed in the anionic yields from photofragmentation of small molecules to provide details of internal structure.
In nuclear physics the concept of "Shape Resonance" is described by Amos de Shalit and Herman Feshbach
in their book.
"It is well known that the scattering from a potential shows characteristics peaks, as a function of energy, for such values of E that make the integral number of wave lengths sit within the potential. The resulting shape resonances are rather broad, their width being of the order of ...."
The shape resonances were observed around the years 1949-1954 in nuclear scattering experiments. They indicate broad asymmetric peaks in the scattering cross section of neutrons or protons scattered by nuclei. The name "shape resonance" has been introduced to describe the fact that the resonance in the potential scattering for the particle of energy E is controlled by the shape of the nucleus. In fact the shape resonance occurs where the integral number of wave lengths of the particle sit within the potential of the nucleus of radius R. Therefore the measure of the energies of the shape resonances in the neutron-nucleus scattering have been used in the years from 1947 to 1954 to measure the radii R of the nuclei with the precision of ±1 x10-13 cm as it can be seen in the chapter "Elastic Cross Sections" of a textbook in Nuclear Physics by R.D. Evans.
The "shape resonances" are discussed in general introductory academic courses of quantum mechanics in the frame of potential scattering phenomena.
The shape resonances arise from the quantum interference between closed and an open scattering channels. At the resonance energy a quasi bound state is degenerate with a continuum. This quantum interference in many body system has been described using quantum mechanics by Gregor Wentzel
, for the interpretation of the Auger effect, by Ettore Majorana
for the dissociation processes and quasi bound states, by Ugo Fano
for the atomic auto-ionization states in the continuum of helium atomic spectrum and by Victor Frederick Weisskopf
. J. M. Blatt and Herman Feshbach
for nuclear scattering experiments.
The shape resonances are related with the existence of nearly stable bound states (that is, resonances) of two objects that dramatically influences how those two objects interact when their total energy is near that of the bound state. When the total energy of the objects is close to the energy of the resonance they interact strongly, and their scattering cross-section becomes very large.
A particular type of "shape resonance" occurs in multiband or two-band superconducting heterostructures at atomic limit called superstripes
due to quantum interference of a first pairing channel in a first wide band and a second pairing channel in a second band where the chemical poteintial is tuned near a Lifshitz transition at the band edge or at the topological electronic transitions of the Fermi surface type "neck-collapsing" or "neck-disrupting"
Serhan describes a state as being a shape resonance if, "the internal state of the system remains unchanged upon disintegration of the quasi-bound level."
A more general discussion of resonances and their taxonomies in molecular system can be found in the review article by Schulz,; for the discovery of the Fano resonance
Fano resonance
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...
line-shape and for the 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...
pioneering work in this field by Antonio Bianconi; and for
a mathematical review by Combes et al..
Quantum mechanics
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a shape resonance, in contrast to a Feshbach resonance
Feshbach resonance
In physics, Feshbach resonance, named after Herman Feshbach, is a resonance of a many-body system in which a bound state is achieved if the coupling between an internal degree of freedom and the reaction coordinates which lead to dissociation vanish...
, is a resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
which is not turned into a bound state
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
if the coupling between some degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
and the degrees of freedom associated to the fragmentation (reaction coordinate
Reaction coordinate
In chemistry, a reaction coordinate is an abstract one-dimensional coordinate which represents progress along a reaction pathway. It is usually a geometric parameter that changes during the conversion of one or more molecular entities....
s) are set to zero. More simply, the shape resonance total energy is more than the separated fragment energy.
Practical implications of this difference for lifetimes and spectral widths are mentioned in works such as Zobel.
Related terms include a special kind of shape resonance, the core-excited shape resonance
Core-excited shape resonance
A core-excited shape resonance is a shape resonance in a system with more than one degree of freedom where, after fragmentation, one of the fragments is in an excited state. It is sometimes very difficult to distinguish a core-excited shape resonance from a Feshbach resonance.-External links:*...
, and trap-induced shape resonance.
Of course in one-dimensional systems, resonances are shape resonances. In a system with more than one degree of freedom, this definition makes sense only if the separable model, which supposes the two groups of degrees of freedom uncoupled, is a meaningful approximation. When the coupling becomes large, the situation is much less clear.
In the case of atomic and molecular electronic structure problems, it is well known that the self-consistent field
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
(SCF) approximation is relevant at least as a starting point of more elaborate methods. The Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
s built from SCF orbitals (atomic
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
or molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s) are shape resonances if only one electronic transition is required to emit one electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
.
Today, there is some debate about the definition and even existence of the shape resonance in some systems observed with molecular spectroscopy. It has been experimentally observed in the anionic yields from photofragmentation of small molecules to provide details of internal structure.
In nuclear physics the concept of "Shape Resonance" is described by Amos de Shalit and Herman Feshbach
Herman Feshbach
Herman Feshbach was an American physicist. He was an Institute Professor Emeritus of physics at MIT. Feshbach is best known for Feshbach resonance and for writing, with Philip M...
in their book.
"It is well known that the scattering from a potential shows characteristics peaks, as a function of energy, for such values of E that make the integral number of wave lengths sit within the potential. The resulting shape resonances are rather broad, their width being of the order of ...."
The shape resonances were observed around the years 1949-1954 in nuclear scattering experiments. They indicate broad asymmetric peaks in the scattering cross section of neutrons or protons scattered by nuclei. The name "shape resonance" has been introduced to describe the fact that the resonance in the potential scattering for the particle of energy E is controlled by the shape of the nucleus. In fact the shape resonance occurs where the integral number of wave lengths of the particle sit within the potential of the nucleus of radius R. Therefore the measure of the energies of the shape resonances in the neutron-nucleus scattering have been used in the years from 1947 to 1954 to measure the radii R of the nuclei with the precision of ±1 x10-13 cm as it can be seen in the chapter "Elastic Cross Sections" of a textbook in Nuclear Physics by R.D. Evans.
The "shape resonances" are discussed in general introductory academic courses of quantum mechanics in the frame of potential scattering phenomena.
The shape resonances arise from the quantum interference between closed and an open scattering channels. At the resonance energy a quasi bound state is degenerate with a continuum. This quantum interference in many body system has been described using quantum mechanics by Gregor Wentzel
Gregor Wentzel
Gregor Wentzel was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel–Kramers–Brillouin approximation in 1926...
, for the interpretation of the Auger effect, by 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...
for the dissociation processes and quasi bound states, by 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...
for the atomic auto-ionization states in the continuum of helium atomic spectrum and by Victor Frederick Weisskopf
Victor Frederick Weisskopf
Victor Frederick Weisskopf was an Austrian-born Jewish American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli and Niels Bohr...
. J. M. Blatt and Herman Feshbach
Herman Feshbach
Herman Feshbach was an American physicist. He was an Institute Professor Emeritus of physics at MIT. Feshbach is best known for Feshbach resonance and for writing, with Philip M...
for nuclear scattering experiments.
The shape resonances are related with the existence of nearly stable bound states (that is, resonances) of two objects that dramatically influences how those two objects interact when their total energy is near that of the bound state. When the total energy of the objects is close to the energy of the resonance they interact strongly, and their scattering cross-section becomes very large.
A particular type of "shape resonance" occurs in multiband or two-band superconducting heterostructures at atomic limit called superstripes
Superstripes
Superstripes are metallic heterostructures at the atomic limit where the shape resonance in the energy gap parameters ∆n is the driving mechanism for the amplification of the superconductivity critical temperature...
due to quantum interference of a first pairing channel in a first wide band and a second pairing channel in a second band where the chemical poteintial is tuned near a Lifshitz transition at the band edge or at the topological electronic transitions of the Fermi surface type "neck-collapsing" or "neck-disrupting"