The term R-matrix has several meanings, depending on the field of study. Its original use has been to mathematically describe
nuclear reactions
In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation o ...
. In particular the general problem of nuclear reactions is to relate the values of the
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
or collision matrix elements (which in principle can be obtained from measurements) to the (slow) dynamics of
nuclear structure
Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics.
Models The cluster model
The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups (e.g., alpha particl ...
. The R-matrix formalism describes the effects of the interaction of the nucleus with the outside world. Its interior is not specified, i.e. it is considered a "black box". The original formulations of the theory came from nuclear scientists
Wigner,
Eisenbud,
Breit,
Blatt,
Weisskopf, and others. Related theories are U-matrix,
S-matrix
In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
, by
M-matrix, or
T-matrix.
Other meanings and uses
The term ''R-matrix'' is used in connection with the
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their ...
, first introduced in the field of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
in the works of J. B. McGuire in 1964 and
C. N. Yang in 1967 and in the
group algebra of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
in the work of
A. A. Jucys in 1966. The classical R-matrix arises in the definition of the classical Yang–Baxter equation.
In
quasitriangular Hopf algebra
In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of H \otimes H such that
:*R \ \Delta(x)R^ = (T \circ \Delta)(x) for all x \in H, where \Delta is the cop ...
, the R-matrix is a solution of the
Yang–Baxter equation.
The numerical modeling of diffraction gratings in
optical science
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultravio ...
can be performed using the R-matrix propagation algorithm.
R-matrix method in quantum mechanics
There is a method in computational
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
for studying scattering known as the R-matrix. Using the original R-matrix theory as a basis, a method was developed for
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
,
positron
The positron or antielectron is the particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
and
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
scattering by
atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s. This approach was later adapted for electron, positron and photon scattering by
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s.
Other applications
R-matrix method is used in
UKRmol and UKRmol+
code suits. The user-friendly software
Quantemol Electron Collisions (Quantemol-EC) and its predecessor
Quantemol-N are based on UKRmol/UKRmol+ and employ
MOLPRO package for electron configuration calculations.
See also
*
Compound nucleus
*
Distorted Waves Born Approximation
*
Optical model
References
{{Reflist, 3
Matrices (mathematics)
Nuclear physics