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 the ...
, 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 based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i ...
undergoing
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...
. (β-decay) 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 during the process of beta decay. There are two forms of beta decay, β ...
and a corresponding
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
, transforming the original
nuclide
A nuclide (or nucleide, from nucleus, also known as nuclear species) is 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 Truman ...
into one with the same mass, but differing charge. (an
isobar)
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 (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov ( uk, Георгій Антонович Гамов, russian: Георгий Антонович Гамов), was a Russian-born Soviet and American polymath, theoreti ...
and
Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care fo ...
at
George Washington University
, mottoeng = "God is Our Trust"
, established =
, type = Private federally chartered research university
, academic_affiliations =
, endowment = $2.8 billion (2022)
, preside ...
.
The weak interaction and beta decay
β 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 Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
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 ...
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 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. Sudarshan has been credited with numerous contrib ...
and
Robert Marshak
Robert Eugene Marshak (October 11, 1916 – December 23, 1992) was an American physicist, educator, and eighth president of the City College of New York.
Biography
Marshak was born in the Bronx, New York City. His parents, Harry and Rose Marshak ...
, and also independently
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
and
Murray Gell-Mann
Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
, determined the correct
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
structure (
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
minus
axial vector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
, ) 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 four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
was developed, which described the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, ...
in terms of massive
gauge bosons
In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge ...
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 of N
Gamow–Teller transition
In nuclear transitions governed by
strong and
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
interactions (which are invariant under
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
), 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
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
and a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
is not meaningful. However, the
weak force
Weak may refer to:
Songs
* "Weak" (AJR song), 2016
* "Weak" (Melanie C song), 2011
* "Weak" (SWV song), 1993
* "Weak" (Skunk Anansie song), 1995
* "Weak", a song by Seether from '' Seether: 2002-2013''
Television episodes
* "Weak" (''Fear t ...
, which governs
beta decay
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...
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 from ...
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 is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
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.
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 isotopes of two different elements where the number of protons of isotope one (Z1) equals the number of neutrons of isotope two (N2) and the number of protons of isotope two (Z2) equals the number of neutr ...
", 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 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 ...
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 al ...
.
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 Nuclear 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 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 ...
says that the transition rate
is given by a transition matrix element (or "amplitude")
weighted by the phase space and Planck's 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
External links
Fermi Theory of Beta Decay
Nuclear physics
Quantum mechanics
Radioactivity