A shape resonance is a metastable state in which an electron is trapped due to the shape of a potential barrier. Altunata 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 Majorana may refer to: * Majorana (surname), an Italian surname * MAJORANA, a physics search for neutrinoless double-beta decay * Majorana fermion * Majorana Prize, a prize for theoretical and mathematical physics See also * Maiorana, a surna ...

pioneering work in this field by Antonio Bianconi; and for a mathematical review by Combes et al.

Quantum mechanics

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

, a shape resonance, in contrast to a Feshbach resonance, is a

resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...

which is not turned into a

bound state Bound or bounds may refer to: Mathematics * Bound variable * Upper and lower bounds, observed limits of mathematical functions Physics * Bound state, a particle that has a tendency to remain localized in one or more regions of space Geography * ...

if the coupling between some

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

and the degrees of freedom associated to the fragmentation (

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. In molec ...

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 In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...

s built from SCF orbitals ( atomic or

molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and 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 findi ...

s) are shape resonances if only one electronic transition is required to emit one

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...

. 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 (February 2, 1917, in New York City – 22 December 2000, in Cambridge, Massachusetts) was an American physicist. He was an Institute Professor Emeritus of physics at MIT. Feshbach is best known for Feshbach resonance and for w ...

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 1949–54 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 wavelengths 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 ±1×10⁻¹³ 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 (17 February 1898 – 12 August 1978) was a German physicist known for development of quantum mechanics. Wentzel, Hendrik Kramers, and Léon Brillouin developed the Wentzel–Kramers–Brillouin approximation in 1926. In his earl ...

, for the interpretation of the Auger effect, by

Ettore Majorana Ettore Majorana (,, uploaded 19 April 2013, retrieved 14 December 2019 ; born on 5 August 1906 – possibly dying after 1959) was an Italian theoretical physicist who worked on neutrino masses. On 25 March 1938, he disappeared under mysteri ...

for the dissociation processes and quasi bound states, by

Ugo Fano Ugo Fano (July 28, 1912 – February 13, 2001) was an Italian American physicist, notable for contributions to theoretical physics. Biography Ugo Fano was born into a wealthy Jewish family in Turin, Italy. His father was Gino Fano, a profes ...

for the atomic auto-ionization states in the continuum of helium atomic spectrum and by

Victor Frederick Weisskopf Victor Frederick "Viki" Weisskopf (also spelled Viktor; September 19, 1908 – April 22, 2002) was an Austrian-born American theoretical physicist. He did postdoctoral work with Werner Heisenberg, Erwin Schrödinger, Wolfgang Pauli, and Niels Boh ...

. J. M. Blatt and

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 is a generic name for a phase with spatial broken symmetry that favors the onset of superconducting or superfluid quantum order. This scenario emerged in the 1990s when non-homogeneous metallic heterostructures at the atomic limit wi ...

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 potential 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"

References

{{DEFAULTSORT:Shape Resonance Scattering Spectroscopy