Editing Beta decay (section)

===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>