In
nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies th ...
, a beta decay transition is the change in state of an
atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
undergoing
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron), transforming into an isobar of that nuclide. For example, beta decay of a neutron ...
. When undergoing beta decay, a nucleus emits a
beta particle
A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus, known as beta decay. There are two forms of beta decay, β− decay and � ...
and a corresponding
neutrino
A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
, transforming the original
nuclide
Nuclides (or nucleides, from nucleus, also known as nuclear species) are a class of atoms characterized by their number of protons, ''Z'', their number of neutrons, ''N'', and their nuclear energy state.
The word ''nuclide'' was coined by the A ...
into one with the same
mass number
The mass number (symbol ''A'', from the German word: ''Atomgewicht'', "atomic weight"), also called atomic mass number or nucleon number, is the total number of protons and neutrons (together known as nucleons) in an atomic nucleus. It is appro ...
but differing
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
(nuclear charge).
There are several types of beta decay transition. In a ''Fermi transition'', the spins of the two emitted particles are anti-parallel, for a combined spin
. As a result, the total angular momentum of the nucleus is unchanged by the transition. By contrast, in a ''Gamow-Teller'' transition, the spins of the two emitted particles are parallel, with total spin
, leading to a change in angular momentum between the initial and final states of the nucleus.
The theoretical work in describing these transitions was done between 1934 and 1936 by
George Gamow
George Gamow (sometimes Gammoff; born Georgiy Antonovich Gamov; ; 4 March 1904 – 19 August 1968) was a Soviet and American polymath, theoretical physicist and cosmologist. He was an early advocate and developer of Georges Lemaître's Big Ba ...
and
Edward Teller
Edward Teller (; January 15, 1908 – September 9, 2003) was a Hungarian and American Theoretical physics, theoretical physicist and chemical engineer who is known colloquially as "the father of the hydrogen bomb" and one of the creators of ...
at
George Washington University
The George Washington University (GW or GWU) is a Private university, private University charter#Federal, federally-chartered research university in Washington, D.C., United States. Originally named Columbian College, it was chartered in 1821 by ...
.
Weak interaction and beta decay
Beta decay had been first described theoretically by
Fermi's original
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural ansatzes or, from German, ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be ...
which was Lorentz-invariant and involved a 4-point fermion vector current. However, this did not incorporate parity violation within the matrix element in
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a ...
seen in weak interactions. The Gamow–Teller theory was necessary for the inclusion of parity violation by modifying the matrix element to include vector and axial-vector couplings of fermions.
This formed the matrix element that completed the Fermi theory of β decay and described parity violation, neutrino helicity, muon decay properties along with the concept of lepton universality. Before the
Standard Model of Particle Physics
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions – excluding gravity) in the universe and classifying all known elementary particles. It ...
was developed,
George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Prof.Sudarshan has been credited with numerous co ...
and
Robert Marshak, and also independently
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
and
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the funda ...
, determined the correct
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
structure (
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
minus
axial vector
Axial may refer to:
* one of the Anatomical terms of location#Other directional terms, anatomical directions describing relationships in an animal body
* In geometry:
:* a geometric term of location
:* an axis of rotation
* In chemistry, referring ...
, ) of the four-fermion interaction.
From there modern
electroweak theory
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forc ...
was developed, which described the
weak interaction
In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
in terms of massive
gauge bosons which was required for describing high energy particle cross-sections.
Fermi transition
In the Fermi transition, the electron and neutrino emitted from the β-decay parent nucleus have spin vectors which are anti-parallel to one another.
This means
:
no change in the total angular momentum of the nucleus
; Examples :
:
:
also
parity is conserved:
.
:
=
excited state
In 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, Add ...
of N
Gamow–Teller transition
In nuclear transitions governed by
strong
Strong may refer to:
Education
* The Strong, an educational institution in Rochester, New York, United States
* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United ...
and
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
interactions (which are invariant under
parity), the physical laws would be the same if the interaction was reflected in a mirror. Hence the sum of a
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
and a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ' ...
is not meaningful. However, the
weak interaction
In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
, which governs
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron), transforming into an isobar of that nuclide. For example, beta decay of a neutron ...
and the corresponding nuclear transitions, ''does'' depend on the
chirality
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable fro ...
of the interaction, and in this case pseudovectors and vectors ''are'' added.
The Gamow–Teller transition is a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ' ...
transition, that is, the selection rules for beta decay caused by such a transition involve no parity change of the nuclear state. The spin of the parent nucleus can either remain unchanged or change by ±1. However, unlike the Fermi transition, transitions from spin 0 to spin 0 are excluded.
In terms of total nuclear angular momentum, the Gamow–Teller transition (
) is
:
; Examples :
; :
; :
also
parity is conserved:
the final
6Li 1
+ state has
and the
state has
states that couple to an even parity state.
Mixed Fermi and Gamow–Teller decay
Due to the existence of the 2 possible final states, each β decay is a mixture of the two decay types. This essentially means that some of the time the remaining nucleus is in an excited state and other times the decay is directly to the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state ...
.
Unlike Fermi transitions, Gamow–Teller transitions occur via an operator that operates only if the initial nuclear wavefunction and final nuclear wavefunction are defined.
The Isospin and Angular Momentum selection rules can be deduced from the operator and the identification of allowed and forbidden decays can be found.
; Examples :
; :
; :
or
; :
; :
The above reaction involves "
mirror nuclei
In physics, mirror nuclei are a pair of isobars of two different elements where the number of protons of isobar one (Z1) equals the number of neutrons of isobar two (N2) and the number of protons of isotope two (Z2) equals the number of neutron ...
", nuclei in which the numbers of protons and neutrons are interchanged.
One can measure the angular distributions of β particles with respect to the axis of nuclear
spin polarization
In particle physics, spin polarization is the degree to which the spin, i.e., the intrinsic angular momentum of elementary particles, is aligned with a given direction. This property may pertain to the spin, hence to the magnetic moment, of co ...
to determine what the mixture is between the two decay types (Fermi and Gamow–Teller).
The mixture can be expressed as a ratio of matrix elements (
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a ...
relates transitions to matrix elements)
:
The interesting observation is that ''y'' for mirror nuclei is on the order of the value of ''y'' for neutron decay while non-mirror nuclear decays tend to be an order of magnitude less.
Physical consequences
Conservation of weak vector current
The Conservation of Vector Current hypothesis was created out of the Gamow–Teller theory. The Fermi decay is the result of a vector current and is dominant in the decay of the neutron to a proton while the Gamow–Teller decay is an axial-current transition. Conservation of Vector Current is the assumption that the weak vector current responsible for the decay is conserved. Another observation is that the Fermi transitions illustrate how the nucleons inside the nucleus interact as free particles despite being surrounded by mesons mediating the nuclear force. This is useful in considering the barrier tunnelling mechanism involved with alpha decay and in deriving the
Geiger–Nuttall law
In nuclear physics, the Geiger–Nuttall law or Geiger–Nuttall rule relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. Roughly speaking, it states that short-lived isotopes emit more energetic alp ...
.
Forbidden decays
The Fermi decays (
) are often referred to as the "superallowed" decays while Gamow–Teller (
) decays are simple "allowed" decays.
Forbidden decays are those which are substantially more improbable, due to parity violation, and as a result have long decay times.
Now the angular momentum (''L'') of the
systems can be non-zero (in the center-of-mass frame of the system).
Below are the observed selection rules for beta decay:
Each of the above have Fermi (
) and Gamow–Teller (
) decays.
So for the "first-forbidden" transitions you have
:
Fermi
and
:
Gamow–Teller
systems.
Notice that
(parity violating transition).
The half life of the decay increases with each order:
:
Decay rate
A calculation of the β emission decay rate is quite different from a calculation of α decay. In α decay the nucleons of the original nucleus are used to form the final state α particle (
4He). In β decay the β and neutrino particles are the result of a nucleon transformation into its isospin complement ( or ). Below is a list of the differences:
# The β electron and neutrino did not exist before the decay.
# The β electron and neutrino are relativistic (nuclear decay energy is usually not enough to make the heavy α nucleus relativistic).
# The light decay products can have continuous energy distributions (before, assuming the α carried away most of the energy was usually a good approximation).
The β decay rate calculation was developed by Fermi in 1934 and was based on Pauli's neutrino hypothesis.
Fermi's Golden Rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a ...
says that the transition rate
is given by a transition matrix element (or "amplitude")
weighted by the phase space and the reduced Planck constant
such that
:
From this analysis we can conclude that the Gamow–Teller nuclear transition from 0 → ±1 is a weak perturbation of the system's interaction
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. This assumption appears to be true based on the very short time scale (10
−20 s) it takes for the formation of quasi-stationary nuclear states compared with the time it takes for a β decay (half lives ranging from seconds to days).
The matrix element between parent and daughter nuclei in such a transition is:
:
with the interaction Hamiltonian forming 2 separate states from the perturbation.
[
]
:
References
{{reflist
External links
Fermi Theory of Beta Decay
Nuclear physics
Quantum mechanics
Radioactivity
George Gamow