Editing Beta decay (section)

==Energy release==
The [[Q value (nuclear science)|{{mvar|Q}} value]] is defined as the total energy released in a given nuclear decay. In beta decay, {{mvar|Q}} is therefore also the sum of the kinetic energies of the emitted beta particle, neutrino, and recoiling nucleus. (Because of the large mass of the nucleus compared to that of the beta particle and neutrino, the kinetic energy of the recoiling nucleus can generally be neglected.) Beta particles can therefore be emitted with any [[kinetic energy]] ranging from 0 to {{mvar|Q}}.<ref name="konya74"/> A typical {{mvar|Q}} is around 1&nbsp;[[MeV]], but can range from a few [[keV]] to a few tens of MeV.

Since the [[Mass–energy equivalence|rest mass]] of the electron is 511&nbsp;keV, the most energetic beta particles are [[Ultrarelativistic limit|ultrarelativistic]], with speeds very close to the [[speed of light]].
In the case of {{sup|187}}Re, the maximum speed of the beta particle is only 9.8% of the speed of light.

The following table gives some examples:

{| class="wikitable zebra"
|+ Examples of beta decay energies
|-
! Isotope || Energy<br />([[Electronvolt|keV]])|| Decay mode 
|-
| free<br />[[neutron]] || {{0}}782.33 ||style="text-align:center;"| β<sup>−</sup> 
|-
| <sup>{{0|00}}3</sup>H<br /><small>([[tritium]])</small> || {{0|00}}18.59 || align="center"| β<sup>−</sup> 
|-
| [[carbon-11|<sup>{{0}}11</sup>C]] || {{0}}960.4<br />1982.4 ||style="text-align:center;"| β<sup>+</sup><br /><span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> 
|-
| [[Carbon-14|<sup>{{0}}14</sup>C]] || {{0}}156.475 ||style="text-align:center;"| β<sup>−</sup> 
|-
| [[fluorine-20|<sup>{{0}}20</sup>F]] || 5390.86 ||style="text-align:center;"| β<sup>−</sup> 
|-
| [[Potassium-37|<sup>{{0}}37</sup>K]] || 5125.48<br />6147.48 ||style="text-align:center;"| β<sup>+</sup><br /><span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> 
|-
| [[Holmium-163|<sup>163</sup>Ho]] || {{0|000}}2.555 ||style="text-align:center;"| <span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> 
|-
| [[Rhenium-187|<sup>187</sup>Re]] || {{0|000}}2.467 ||style="text-align:center;"| β<sup>−</sup> 
|-
|[[Bismuth-210|<sup>210</sup>Bi]] || 1162.2 ||style="text-align:center;"| β<sup>−</sup> 
|}
Tritium β<sup>−</sup> decay being used in the [[KATRIN]] experimental search for [[sterile neutrinos]].<ref>{{Cite journal |last=Mertens |first=Susanne |date=2015-01-01 |title=Status of the KATRIN Experiment and Prospects to Search for keV-mass Sterile Neutrinos in Tritium β-decay |journal=Physics Procedia |series=13th International Conference on Topics in Astroparticle and Underground Physics, TAUP 2013 |volume=61 |pages=267–273 |doi=10.1016/j.phpro.2014.12.043 |issn=1875-3892|doi-access=free |bibcode=2015PhPro..61..267M }}</ref>
===β<sup>−</sup> decay===
Consider the generic equation for beta decay
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''+1}}|X′}} + {{SubatomicParticle|Electron}} + {{math|{{SubatomicParticle|Electron Antineutrino}}}}.
The {{mvar|Q}} value for this decay is
:<math chem>Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m_N\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)-m_e-m_{\overline\nu_e}\right]c^2</math>,
where <math chem>m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)</math> is the mass of the nucleus of the {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} atom, <math chem>m_e</math> is the mass of the electron, and <math chem>m_{\overline\nu_e}</math> is the mass of the electron antineutrino. In other words, the total energy released is the mass energy of the initial nucleus, minus the mass energy of the final nucleus, electron, and antineutrino. The mass of the nucleus {{mvar|m<sub>N</sub>}} is related to the standard [[atomic mass]] {{mvar|m}} by
<math chem display="block">m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)c^2=m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)c^2 + Z m_e c^2-\sum_{i=1}^Z B_i.</math>
That is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of all ''electron'' binding energies {{mvar|B<sub>i</sub>}} for the atom. This equation is rearranged to find <math chem>m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)</math>, and <math chem>m_N\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)</math> is found similarly. Substituting these nuclear masses into the {{math|Q}}-value equation, while neglecting the nearly-zero antineutrino mass and the difference in electron binding energies, which is very small for high-{{mvar|Z}} atoms, we have
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)-m\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)\right]c^2</math>
This energy is carried away as kinetic energy by the electron and antineutrino.

Because the reaction will proceed only when the {{mvar|Q}}&nbsp;value is positive, β<sup>−</sup> decay can occur when the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} is greater than the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''+1}}|X′}}.<ref name="Krane1987">{{cite book|author=Kenneth S. Krane|title=Introductory Nuclear Physics| url=https://books.google.com/books?id=ConwAAAAMAAJ|date=5 November 1987|publisher=Wiley|isbn=978-0-471-80553-3}}</ref>

===β<sup>+</sup> decay===
The equations for β<sup>+</sup> decay are similar, with the generic equation
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{SubatomicParticle|Positron}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}
giving
<math chem display="block">Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m_N\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-m_e-m_{\nu_e}\right]c^2.</math>
However, in this equation, the electron masses do not cancel, and we are left with
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)-m\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-2m_e\right]c^2.</math>

Because the reaction will proceed only when the {{mvar|Q}}&nbsp;value is positive, β<sup>+</sup> decay can occur when the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} exceeds that of {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} by at least twice the mass of the electron.<ref name="Krane1987" />

===Electron capture===
The analogous calculation for electron capture must take into account the binding energy of the electrons. This is because the atom will be left in an excited state after capturing the electron, and the binding energy of the captured innermost electron is significant. Using the generic equation for electron capture
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} + {{SubatomicParticle|Electron}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}
we have
<math chem display="block">Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) + m_e - m_N\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-m_{\nu_e}\right]c^2,</math>
which simplifies to
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)\right]c^2-B_n,</math>
where {{mvar|B<sub>n</sub>}} is the binding energy of the captured electron.

Because the binding energy of the electron is much less than the mass of the electron, nuclei that can undergo β<sup>+</sup> decay can always also undergo electron capture, but the reverse is not true.<ref name="Krane1987" />