Editing Beta decay (section)

==Beta emission spectrum==
[[File:Beta spectrum of RaE.jpg|thumb|Beta spectrum of <sup>210</sup>Bi. ''E''<sub>max</sub> = ''Q'' = 1.16 MeV is the maximum energy]]
Beta decay can be considered as a [[Perturbation theory (quantum mechanics)|perturbation]] as described in quantum mechanics, and thus [[Fermi's Golden Rule]] can be applied. This leads to an expression for the kinetic energy spectrum {{math|''N''(''T'')}} of emitted betas as follows:<ref>{{cite web | last=Nave | first=C. R. |title=Energy and Momentum Spectra for Beta Decay | url=http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta2.html |work=[[HyperPhysics]] |access-date=2013-03-09}}</ref>

<math display="block">N(T) = C_L(T) F(Z,T) p E (Q-T)^2</math>

where {{mvar|T}} is the kinetic energy, {{mvar|C<sub>L</sub>}} is a shape function that depends on the forbiddenness of the decay (it is constant for allowed decays), {{math|''F''(''Z'', ''T'')}} is the Fermi Function (see below) with ''Z'' the charge of the final-state nucleus, {{math|1=''E'' = ''T'' + ''mc''<sup>2</sup>}} is the total energy, <math> p = \sqrt{(E/c)^2 - (mc)^2}</math> is the momentum, and {{mvar|Q}} is the [[Q value (nuclear science)|Q value]] of the decay. The kinetic energy of the emitted neutrino is given approximately by {{mvar|Q}} minus the kinetic energy of the beta.

As an example, the beta decay spectrum of <sup>210</sup>Bi (originally called RaE) is shown to the right.

===Fermi function===

The Fermi function that appears in the beta spectrum formula accounts for the Coulomb attraction / repulsion between the emitted beta and the final state nucleus. Approximating the associated wavefunctions to be spherically symmetric, the Fermi function can be analytically calculated to be:<ref>{{cite journal |last1=Fermi |first1=E. |year=1934 |title=Versuch einer Theorie der &beta;-Strahlen. I |journal=[[Zeitschrift für Physik]] |volume=88 |issue=3–4 |pages=161–177 |bibcode=1934ZPhy...88..161F |doi=10.1007/BF01351864|s2cid=125763380 }}</ref>

<math display="block">F(Z,T)=\frac{2 (1+S)}{\Gamma(1+2S)^2} (2 p \rho)^{2S-2} e^{\pi \eta} |\Gamma(S+i \eta)|^2,</math>

where {{mvar|p}} is the final momentum, Γ the [[Gamma function]], and (if {{mvar|α}} is the [[fine-structure constant]] and {{mvar|r<sub>N</sub>}} the radius of the final state nucleus) <math>S = \sqrt{1 - \alpha^2 Z^2}</math>, <math>\eta = \pm Ze^2E/(\hbar cp)</math> (+ for electrons, − for positrons), and <math>\rho = r_N/\hbar </math>.

For non-relativistic betas ({{math|''Q'' ≪ ''m''<sub>e</sub>''c''<sup>2</sup>}}), this expression can be approximated by:<ref>{{cite book
 |last1=Mott |first1=N. F.
 |last2=Massey |first2=H. S. W.
 |year=1933
 |title=The Theory of Atomic Collisions
 |publisher=[[Clarendon Press]]
 |bibcode=1933tac..book.....M
 |lccn=34001940
}}</ref>

<math display="block">F(Z,T) \approx \frac{2 \pi \eta}{1 - e^{- 2 \pi \eta}}.</math>

Other approximations can be found in the literature.<ref>{{cite journal
 |last1=Venkataramaiah |first1=P.
 |last2=Gopala |first2=K.
 |last3=Basavaraju |first3=A.
 |last4=Suryanarayana |first4=S. S.
 |last5=Sanjeeviah |first5=H.
 |year=1985
 |title=A simple relation for the Fermi function
 |journal=[[Journal of Physics G]]
 |volume=11 |issue=3 |pages=359–364
 |bibcode=1985JPhG...11..359V
 |doi=10.1088/0305-4616/11/3/014
|s2cid=250803189
 }}</ref><ref>{{cite journal
 |last1=Schenter |first1=G. K.
 |last2=Vogel |first2=P.
 |year=1983
 |title=A simple approximation of the fermi function in nuclear beta decay
 |journal=[[Nuclear Science and Engineering]]
 |volume=83 |issue=3 |pages=393–396
 |doi=10.13182/NSE83-A17574
 |bibcode=1983NSE....83..393S
 |osti=5307377
}}</ref>

===Kurie plot===
<!-- Note: Kurie is correct (see References below); no relation to the Curies -->

A '''Kurie plot''' (also known as a '''Fermi–Kurie plot''') is a graph used in studying beta decay developed by [[Franz N. D. Kurie]], in which the square root of the number of beta particles whose momentum (or energy) lies within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy.<ref>{{cite journal
 |last1=Kurie |first1=F. N. D. |author-link=Franz N. D. Kurie
 |last2=Richardson |first2=J. R.
 |last3=Paxton |first3=H. C.
 |year=1936
 |title=The Radiations Emitted from Artificially Produced Radioactive Substances. I. The Upper Limits and Shapes of the β-Ray Spectra from Several Elements
 |journal=[[Physical Review]]
 |volume=49 |issue=5 |pages=368–381
 |bibcode=1936PhRv...49..368K
 |doi=10.1103/PhysRev.49.368
}}</ref><ref>{{cite journal
 |last1=Kurie |first1=F. N. D. |author-link=Franz N. D. Kurie
 |year=1948
 |title=On the Use of the Kurie Plot
 |journal=[[Physical Review]]
 |volume=73 |issue=10 |page=1207
 |bibcode=1948PhRv...73.1207K
 |doi=10.1103/PhysRev.73.1207
}}</ref> It is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory. The energy-axis (x-axis) intercept of a Kurie plot corresponds to the maximum energy imparted to the electron/positron (the decay's {{mvar|Q}}&nbsp;value). With a Kurie plot one can find the limit on the effective mass of a neutrino.<ref>{{Cite journal
 |last=Rodejohann |first=W.
 |year=2012
 |title=Neutrinoless double beta decay and neutrino physics
 |journal=Journal of Physics G: Nuclear and Particle Physics
 |volume=39
 |issue=12
 |page=124008
 |arxiv=1206.2560
 |doi=10.1088/0954-3899/39/12/124008
 |bibcode=2012JPhG...39l4008R
 |s2cid=119158221 }}</ref>