Beta decay transition: Difference between revisions

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{{short description|Physical phenomenon}}

{{expert needed|date=August 2016}}

{{expert needed|1=Physics|date=August 2016|reason=This article still needs revision to make it generally about transitions, to simplify and elaborate the discussion to be perhaps less technical to non experts, and to reduce the redundancies with the beta decay page}}

{{technical|date=August 2016}}

A '''Fermi transition''' or a '''Gamow–Teller transition''' are types of [[Beta decay|nuclear beta decay]] determined by changes in angular momentum or spin. In the Fermi transition, the spins of the emitted particles are antiparallel, coupling to <math>S=0</math>, so the angular momentum of the initial and final angular momentum states of the nucleus are unchanged (<math>\Delta J=0</math>). This is in contrast to a Gamow-Teller transition, where the spins of the emitted electron (positron) and antineutrino (neutrino) couple to total spin <math>S=1</math>, leading to an angular momentum change <math>\Delta J = 0,\pm 1</math> between the initial and final angular momentum states of the nucleus.

Fermi and Gamow-Teller transitions correspond to two different forms of leading order behavior of the weak interaction Hamiltonian in the non-relativistic limit:

<ref>

{{cite book

|author=Samuel S.M. Wong

|year=2004

|title=Introductory Nuclear Physics (2nd Edition)

|page=192

}}</ref>

In [[nuclear physics]], a '''beta decay transition''' is the change in state of an [[atomic nucleus]] undergoing [[beta decay]]. When undergoing beta decay, a nucleus emits a [[beta particle]] and a corresponding [[neutrino]], transforming the original [[nuclide]] into one with the same [[mass number]] but differing [[atomic number]] (nuclear charge).

<math>\hat{H}_\text{int} = \begin{cases} G_{V}\hat{1} \hat{\tau} & \text{Fermi decay} \\ G_{A}\hat{\sigma} \hat{\tau} & \text{Gamow–Teller decay} \end{cases}</math>

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 <math>S=0</math>. 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 <math>S=1</math>, leading to a change in angular momentum between the initial and final states of the nucleus.<ref>{{cite book |last1=Clayton |first1=Donald D. |title=Principles of stellar evolution and nucleosynthesis : with a new preface |date=1983 |publisher=University of Chicago Press |___location=Chicago |isbn=0-226-10953-4 |page=366-367 |edition=University of Chicago Press}}</ref>

: <math>\hat{\tau}</math> = isospin transition matrix which turn protons to neutrons and vise-versa

: <math>\hat{\sigma}</math> = [[Pauli matrices|Pauli spin matrices]], which lead to <math>\Delta J = 0,\pm 1</math>.

: <math>\hat{1}</math> = identity operator in spin space, leaving <math> J </math> unchanged.

: <math>G_{V}</math> = Weak vector coupling constant.

: <math>G_{A}</math> = Weak axial-vector coupling constant.

The theoretical work in describing these transitions was done between 1934 and 1936 by Nuclear Physicists [[George Gamow]] and [[Edward Teller]] at [[George Washington University]].

==The weakWeak interaction and beta decay ==

{| class="wikitable" style="float:right; clear:right; margin-top:0; margin-left:10px; margin-bottom:8px; margin-right:0; padding:7px; font-size:85%; width:230px;"

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| {{nowrap|[[File:MuonFermiDecay.gif|x110px]]&nbsp;}}

|-

| [[Fermi's interaction]] showing the 4-point fermion vector current, coupled under the Fermi's coupling constant, "Gf"''G''_F. Fermi's theory was the first theoretical effort in describing nuclear decay rates for [[beta decay]]. The Gamow–Teller theory was a necessary extension of Fermi's theory.

|}

βBeta decay had been first described theoretically by [[Enrico Fermi|Fermi's]]'s original [[ansatz]] 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 Goldengolden Rulerule]] 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| Standard Model of Particle Physics]] was developed, [[George Sudarshan]] and [[Robert Marshak]], and also independently [[Richard Feynman]] and [[Murray Gell-Mann]], determined the correct [[tensor]] structure ([[vector (geometric)|vector]] minus [[axial vector]], {{nowrap|''V'' − ''A''}}) of the four-fermion interaction.

From there modern [[Quantum Field Theory|electroweak theory]] was developed, which described the [[weak interaction]] in terms of massive [[W and Z Bosons|Gaugegauge Bosonsbosons]] which was required for describing high energy particle cross-sections.

== Fermi transition ==

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This means

: <math>\Delta I = 0 \Rightarrow</math> no change in the total angular momentum of the nucleus

; Examples :

: <math>{}^{14}_8 \text{O}_6 \rightarrow {}^{14}_7 \text{N}^*_7 + \beta^+ + \nu_\text{e}</math>

: <math>I_i = 0^+ \rightarrow I_f = 0^+ \Rightarrow \Delta I = 0</math>

also <math>\Delta \pi = 0 \Rightarrow</math> parity is conserved: <math>\pi (Y_{\ell\,m}) =(-1)^{\ell}</math>.

: <math>{}^{14}_7 \text{N}^*_{7}</math> = [[excited state]] of N

:<math>{}^{14}_7 \text{N}^*_{7}</math> = excited state of N

;Examples:

: <math>{}^6_2 \text{He}_4 \rightarrow {}^6_3 \text{Li}_3 + \beta^- + \bar{\nu}_\text{e}</math>

: <math>I_i = 0^+ \rightarrow I_f = 1^+ \Rightarrow \Delta I = 1</math>

also <math>\Delta \pi = 0 \Rightarrow</math> parity is conserved: <math>\pi (Y_{\ell\,m}) =(-1)^{\ell} \Rightarrow</math> the final ⁶Li 1⁺ state has <math>L = 1</math> and the <math>\beta + \bar{\nu}_\text{e}</math> state has <math>S = 1</math> states that couple to an even parity state.

== Gamow–Teller transition ==

In nuclear transitions governed by [[strong interaction|strong]] and [[Electromagnetism|electromagnetic]] interactions (which are invariant under [[parity (physics)|parity]]), the physical laws would be the same if the interaction was reflected in a mirror. Hence the sum of a [[Vector (geometry)|vector]] and a [[pseudovector]] is not meaningful. However, the [[weak forceinteraction]], which governs [[beta decay]] and the corresponding nuclear transitions, ''does'' depend on the [[chirality (physics)|chirality]] of the interaction, and in this case pseudovectors and vectors ''are'' added.

The Gamow–Teller transition is a [[pseudovector]] transition, that is, the selection rules for beta decay caused by such a transition involve no parity change of the nuclear state.<ref>

{{cite journal |title=Nuclear spin and isospin excitations|journal=Reviews of Modern Physics|volume=64|issue=2|pages=491–557 |author=Franz Osterfeld |doi=10.1103/RevModPhys.64.491|year=1992|bibcode=1992RvMP...64..491O}}</ref> 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.

{{cite web

|url=http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.64.491

|title=Nuclear spin and isospin excitations|publisher=Rev. Mod. Phys. 64, 491 (1992) - APS Journals

|author=Franz Osterfeld

}}</ref> 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 (<math>I_i \rightarrow I_f</math>) is

: <math>\Delta I = I_f - I_i = \begin{cases} 0 & I_i = I_f = 0 \\ 1 & I_i = 0 \text{ and } I_f = 1 \end{cases}</math>

; Examples :

:<math>\Delta I = I_f - I_i = \begin{cases} 0 & I_i = I_f = 0 \\ 1 & I_i = 0 \text{ and } I_f = 1 \end{cases}</math>

; : <math>{}^6_2 \text{He}_4 \rightarrow {}^6_3 \text{Li}_3 + \beta^- + \bar{\nu}_\text{e}</math>

; : <math>I_i = 0^+ \rightarrow I_f = 1^+ \Rightarrow \Delta I = 1</math> also <math>\Delta \pi = 0 \Rightarrow</math> parity is conserved: <math>\pi (Y_{\ell\,m}) =(-1)^{\ell} \Rightarrow</math> the final ⁶Li 1⁺ state has <math>L = 1</math> and the <math>\beta + \bar{\nu}_\text{e}</math> state has <math>S = 1</math> 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.<ref>

<ref>

{{cite book

|author=Samuel S.M. Wong

|year=2004

|title=Introductory Nuclear Physics (2nd Edition)

|url=https://archive.org/details/introductorynucl00wong_914

|page=198

|url-access=limited

}}</ref>

|page=[https://archive.org/details/introductorynucl00wong_914/page/n212 198]

|publisher=Wiley-VCH

;Examples:

|edition=2nd

}}</ref>

: <math>{}^{21}_{11} \text{Na}_{10} \rightarrow {}^{21}_{10} \text{Ne}_{11} + \beta^+ + \nu_\text{e}</math>

: <math>I_i = \frac{3}{2}^+ \Rightarrow I_f = \frac{3}{2}^+ \Rightarrow \Delta I = 0</math>

; Examples :

; : <math>{}^{21}_{11} \text{Na}_{10} \rightarrow {}^{21}_{10} \text{Ne}_{11} + \beta^+ + \nu_\text{e}</math>

; : <math>I_i = \frac{3}{2}^+ \Rightarrow I_f = \frac{3}{2}^+ \Rightarrow \Delta I = 0</math>

or

; : <math>{}^{21}_{11} \text{Na}_{10} \rightarrow {}^{21}_{10} \text{Ne}^*_{11} + \beta^+ + \nu_\text{e}</math>

; : <math>{}^{21}_{11}I_i = \textfrac{Na3}_{102}^+ \rightarrowRightarrow {}^{21}_{10}I_f = \textfrac{Ne5}^*_{112} ^+ \beta^+Rightarrow +\Delta \nu_\text{e}I = 1</math>

: <math>I_i = \frac{3}{2}^+ \Rightarrow I_f = \frac{5}{2}^+ \Rightarrow \Delta I = 1</math>

The above reaction involves "[[mirror nuclei]]", 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]] relates transitions to matrix elements)

:<math>y \equiv \frac{g_\text{F} M_\text{F}}{g_\text{GT} M_\text{GT}}</math><ref>{{cite journal |title=The Fermi to Gamow–Teller mixing ratio of the β⁺ decay of ⁵²Mn and time-reversal invariance |date=1988-11-03 |doi=10.1007/BF01295458 |last1=Saw |first1=E. L. |last2=Yap |first2=C. T. |journal=Zeitschrift für Physik A |volume=332 |issue=3 |pages=285–287 |s2cid=120281084 }}</ref>

:<math>y \equiv \frac{g_\text{F} M_\text{F}}{g_\text{GT} M_\text{GT}}</math><ref>

{{cite web

|url=http://www.springerlink.com/content/ju13j4786530u057/

|title=The Fermi to Gamow–Teller mixing ratio of theβ+ decay of52Mn and time-reversal invariance |publisher= E. L. Saw and C. T. Yap

|accessdate=1988-11-3

}}</ref>

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.

Line 129 ⟶ 94:

Now the angular momentum (''L'') of the <math>\beta + \nu</math> systems can be non-zero (in the center-of-mass frame of the system).

Below are the Observedobserved Selectionselection Rulesrules for Nuclearbeta Beta-Decaydecay:<ref>

{{cite book

|author=Samuel S.M. Wong

|year=2004

|title=Introductory Nuclear Physics (2nd Edition)

|url=https://archive.org/details/introductorynucl00wong_914

|page=200

|url-access=limited

}}</ref>

|page=[https://archive.org/details/introductorynucl00wong_914/page/n214 200]

|publisher=Wiley-VCH

|edition=2nd

}}</ref>

{| class="wikitable"

Line 147 ⟶ 116:

| first-forbidden (parity change) || 1 || 0, 1, 2 || 1

|-

| second-forbidden (no parity change) || 2 || 1, 2, 3 || 0

|-

| third-forbidden (parity change) || 3 || 2, 3, 4 || 1

|-

| fourth-forbidden (no parity change) || 4 || 3, 4, 5 || 0

|}

Line 157 ⟶ 126:

So for the "first-forbidden" transitions you have

: <math>\vec{I} = \vec{L} + \vec{S} = \vec{1} + \vec{0} \Rightarrow \Delta I = 0,1</math> Fermi

and

: <math>\vec{I} = \vec{L} + \vec{S} = \vec{1} + \vec{1} \Rightarrow \Delta I = 0,1,2</math> Gamow–Teller

systems.

Line 171 ⟶ 140:

|page=303

|___location=University of California

}}</ref>

: <math>

\begin{align}

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+ \beta^+ + \nu_\text{e} &

t_{1/2} &= 2.6\,\text{years} \\

{}^{115}_{49} \text{In}_{7666} \left(\frac{9}{2}^+\right) &\rightarrow

{}^{115}_{50} \text{Sn}_{7565} \left(\frac{1}{2}^+\right)

+ \beta^- + \bar{\nu}_\text{e} &

t_{1/2} &= 10^{14}\,\text{years}

Line 186 ⟶ 154:

</math>

=== 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 ({{nowrap|n → p}} or {{nowrap|p → n}}). Below is a list of the differences:

# The β electron and neutrino did not exist before the decay.

#the The β electron and neutrino didare relativistic (nuclear decay energy is usually not existenough beforeto make the decayheavy 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 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]] says that the transition rate <math>W</math> is given by a transition matrix element (or "amplitude") <math>M_{i,f}</math> weighted by the phase space and the reduced Planck's constant <math>\hbar</math> such that

: <math>W = \frac{2 \pi}{\hbar} \left| M_{i,f} \right|^2 \times \text{(Phase Space)} = \frac{\ln 2}{t_{1/2}}</math>

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 mechanics#Mathematical formalism|Hamiltonian]]. This assumption appears to be true based on the very short time scale (10⁻²⁰ &nbsp;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:

: <math>\left| M_{i,f} \right|^2 = \left\langle \psi_\text{Daughter} \phi_\beta \psi_\nu \right| \hat{H}_\text{int} \left| \psi_\text{Parent} \right\rangle</math>

with the interaction Hamiltonian forming 2 separate states from the perturbation.<ref>

<math>\left| M_{i,f} \right|^2 = \left\langle \psi_\text{Daughter} \phi_\beta \psi_\nu \right| \hat{H}_\text{int} \left| \psi_\text{Parent} \right\rangle</math>

with the interaction Hamiltonian forming 2 separate states from the perturbation.

<ref>

{{cite book

|author=Samuel S.M. Wong

|year=2004

|title=Introductory Nuclear Physics (2nd Edition)

|url=https://archive.org/details/introductorynucl00wong_914

|page=192

|url-access=limited

}}</ref>

|page=[https://archive.org/details/introductorynucl00wong_914/page/n206 192]

|publisher=Wiley-VCH

<math>\hat{H}_\text{int} = \begin{cases} G_{V}\hat{1} \hat{\tau} & \text{Fermi decay} \\ G_{A}\hat{\sigma} \hat{\tau} & \text{Gamow–Teller Decay} \end{cases}</math>

|edition=2nd

}}</ref>

: <math>\hat{H}_\text{int} = \begin{cases} G_{V}\hat{1} \hat{\tau} & \text{Fermi decay} \\ G_{A}\hat{\sigma} \hat{\tau} & \text{Gamow–Teller Decay} \end{cases}</math>

== References ==

{{reflist}}

<references />

== External links ==

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* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi.html#c1 Transition Probabilities and Fermi's Golden Rule]

{{DEFAULTSORT:Gamow-Teller transition}}

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[[Category:Quantum mechanics]]

[[Category:Radioactivity]]

[[Category:George Gamow]]