Primitive root modulo n

2014-12-21T14:29:53Z

174.102.27.76: /* Examples */ misspelling: expect -> except

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Magnetic dipole transition

2014-11-23T02:20:43Z

174.102.100.9: /* Theoretical description */ edited equation formatting for clarity

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Topological complexity

2013-08-03T15:25:01Z

174.102.111.124: /* References */

{{jargon|date=July 2012}}
{{context|date=July 2012}}
A '''Gamow–Teller transition''' is, fundamentally, a type of [[Beta Decay|β decay]] in which the spin vectors of the released [[electron]] and electron [[neutrino|antineutrino]] are parallel.
In contrast, a '''Fermi transition''' is a β decay in which the spin vectors of the released [[electron]] and electron [[neutrino|antineutrino]] are anti-parallel.

In terms of the interaction [[Hamiltonian (quantum mechanics)|Hamiltonian]], the probability amplitude from parent to daughter nucleus in a beta-decay forms 2 possible final states:
<ref>
{{cite book
|author=Samuel S.M. Wong
|year=2004
|title=Introductory Nuclear Physics (2nd Edition)
|page=192
}}</ref>

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

: <math>\hat{\tau}</math> = isospin transition matrix which turn protons to neutrons and vis-versa
: <math>\hat{\sigma}</math> = rotation matrix which changes parent nucleus parity from 0 → 1.

In effect, Gamow–Teller Transitions are an extension of Fermi's theory of β decay and describes parity violation inherent in the [[weak interaction]] in terms of nuclear physics.

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 Weak 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;"
|-
| {{nowrap|[[File:MuonFermiDecay.gif|x110px]] }}
|-
| [[Fermi's interaction]] showing the 4-point fermion vector current, coupled under Fermi's coupling constant, "Gf". 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.
|}
β decay had been first described theoretically by [[Enrico Fermi|Fermi'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 Golden Rule]] 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|Gauge Bosons]] which was required for describing high energy particle cross-sections.

== Conservation rules for Beta Decay ==

=== Baryon number is conserved===

:<math>B = \frac{n_q - n_{\bar{q}}}{3}</math>
where
: <math>n_q</math> is the number of constituent quarks, and
: <math>n_{\overline{q}}</math> is the number of constituent antiquarks.

Beta decay just changes [[neutron]] to [[proton]] or, in the case of negative beta decay ([[electron capture]]) [[proton]] to [[neutron]] so the number of individual [[quarks]] don't change. It is only the baryon flavour flavor that changes, here labelled as the [[isospin]].

Up and down [[quarks]] have total isospin <math>I = \frac{1}{2}</math> and isospin projections
:<math>I_\text{z} = \begin{cases} \frac{1}{2} & \text{up quark} \\ -\frac{1}{2} & \text{down quark} \end{cases}</math>

All other quarks have ''I'' = 0.

In general
:<math>I_\text{z} = \frac{1}{2} (n_\text{u} - n_\text{d})</math>

===Lepton number is conserved===

::<math>L \equiv n_{\ell} - n_{\bar{\ell}}</math>

so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0.
:<math>\begin{matrix}
& \text{n} & \rightarrow & \text{p} & + & \text{e}^- & + & \bar{\nu}_\text{e} \\
L: & 0 & = & 0 & + & 1 & - & 1 \end{matrix}</math>

=== Angular momentum===

Net angular momentum is zero, hence only spin quantum numbers are considered. only consider spins)

The electron and antineutrino are [[fermions]], spin-1/2 objects, therefore their spins may be either parallel or anti-parallel.

== Transition Mixing ==

Due to quantum superposition of final spin states, all transitions in beta decay are a superposition of Fermi (anti-parallel) and Gamow–Teller (parallel) states. Each final state must be considered in order to distinguish the final state.

== Fermi decay ==

In Fermi decay, the electron and neutrino emitted from the β-decay parent nucleus have spin vectors which are anti-parallel to one another.

This means

: <math>\Delta I = 0 \Rightarrow</math> no change in the isospin 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

;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 (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 force]], which governs [[beta decay]], ''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 [[pseudoscalar]] transition, that is the selection rules for beta decay caused by the 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 (<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>

== 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 exited 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>
{{cite book
|author=Samuel S.M. Wong
|year=2004
|title=Introductory Nuclear Physics (2nd Edition)
|page=198
}}</ref>

;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>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 number of protons and neutrons is just interchange.

One can measure the angular distributions of β particles 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 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.

== 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]].

==== Forbidden decays ====

The Fermi decays (<math>\Delta I = 0</math>) are often referred to as the "superallowed" decays while Gamow–Teller (<math>\Delta I = 1</math>) 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 <math>\beta + \nu</math> systems can be non-zero (in the center-of-mass frame of the system).

Below are the Observed Selection Rules for Nuclear Beta-Decay:<ref>
{{cite book
|author=Samuel S.M. Wong
|year=2004
|title=Introductory Nuclear Physics (2nd Edition)
|page=200
}}</ref>

{| class="wikitable"
|-
! Transition !! ''L'' !! Δ''I'' !! Δ''π''
|-
| Fermi || 0 || 0 || 0
|-
| Gamow–Teller || 0 || 0, 1 || 0
|-
| first-forbidden (parity change) || 1 || 0, 1, 2 || 1
|-
| second-forbidden (no parity change) || 2 || 2, 3 || 0
|-
| third-forbidden (parity change) || 3 || 3, 4 || 1
|-
| fourth-forbidden (no parity change) || 4 || 4, 5 || 0
|}

Each of the above have Fermi (<math>S = 0</math>) and Gamow–Teller (<math>S = 1</math>) decays.

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.

Notice that <math>\Delta \pi = 1 \Rightarrow</math> (parity violating transition).

The half life of the decay increases with each order:<ref>
{{cite book
|author=Willard F. Libby
|year=1981
|title=Radioactivity & particle physics and, Radioactive fallout & technology
|page=303
|location=University of California
|unused_data=Location:University of California
}}</ref>

: <math>
\begin{align}
{}^{22}_{11} \text{Na}_{11} \left(3^+\right) &\rightarrow
{}^{22}_{10} \text{Ne}_{12} \left(2^+\right)
+ \beta^+ + \nu_\text{e} &
t_{1/2} &= 2.6\,\text{years} \\
{}^{115}_{49} \text{In}_{76} \left(\frac{9}{2}^+\right) &\rightarrow
{}^{115}_{50} \text{Sn}_{75} \left(\frac{1}{2}^+\right)
+ \beta^- + \bar{\nu}_\text{e} &
t_{1/2} &= 10^{14}\,\text{years}
\end{align}
</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 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 eergy 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 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 purturbation 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 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>
{{cite book
|author=Samuel S.M. Wong
|year=2004
|title=Introductory Nuclear Physics (2nd Edition)
|page=192
}}</ref>

<math>\hat{H}_\text{int} = \begin{cases} \hat{1} \hat{\tau} & \text{Fermi decay} \\ \hat{\sigma} \hat{\tau} & \text{Gamow–Teller Decay} \end{cases}</math>
: <math>\hat{\tau}</math> = isospin transition matrix which turn protons to neutrons and vis-versa
: <math>\hat{\sigma}</math> = rotation matrix which changes parent nucleus parity from 0 → 1.

Hence the Gamow–Teller transition describes parity violation in terms of Fermi's theory of the [[weak interaction]], explaining why certain decays are longer than others or restricted altogether.

== References ==
<references />

== External links ==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi2.html Fermi Theory of Beta Decay ]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi.html#c1 Transition Probabilities and Fermi's Golden Rule]



{{DEFAULTSORT:Gamow-Teller transition}}
[[Category:Nuclear physics]]
[[Category:Quantum mechanics]]
[[Category:Radioactivity]]

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Primitive root modulo n

Magnetic dipole transition

Topological complexity