= Solution
At energies far below the $W$ <mass>, replace its propagator by a constant. The modern <V-A interaction> form of Fermi's contact theory has, in $\hbar=c=1$ units,
$$
\mathcal H_{\rm eff}=\frac{G_FV_{ud}}{\sqrt2}
[\bar\psi_p\gamma^\mu(g_V-g_A\gamma^5)\psi_n]
[\bar\psi_e\gamma_\mu(1-\gamma^5)\psi_\nu]+\mathrm{h.c.}
$$
Nucleons are treated nonrelativistically within nuclear states. To leading order in their velocity, the vector charge and axial spatial current give the two nuclear operators
$$
M_F=\langle f|\sum_k\tau_k^+|i\rangle,\qquad
\mathbf M_{GT}=\langle f|\sum_k\boldsymbol\sigma_k\tau_k^+|i\rangle,
$$
where $\tau^+$ here means neutron-to-proton conversion. \b[(i)] The scalar Fermi operator permits $\Delta J=0$, including $0\to0$, without <parity> change. The rank-one Gamow–Teller operator permits $\Delta J=0,1$ without <parity> change, excluding $0\to0$. Both terms can contribute when the same nuclear states permit them; their strengths contain nuclear matrix elements, not just <phase space>.
\b[(ii)] Expand the <lepton> waves across the <nucleus> in powers of their momenta times nuclear radius. When $pR/\hbar\ll1$, the constant spatial term gives allowed transitions. If its matrix element vanishes by <spin>/<parity> selection, higher terms supply <lepton> <orbital angular momentum>: “forbidden” means suppressed, not impossible. First-forbidden terms change <parity> and can have $\Delta J=0,1,2$; the rank-two $\Delta J=2$ case is unique first-forbidden. Higher orders carry additional small spatial factors and different energy-dependent shape factors. The contact approximation alone does not imply the allowed approximation.
For a single branch, neglect recoil and <neutrino> <mass> and first assume an energy-independent nuclear shape factor. <Fermi's golden rule> integrates the <neutrino> <phase space> with $E_\nu=Q-T$, while the <electron> <momentum> volume is proportional to $p^2dp$. Including Coulomb distortion,
$$
N(p)\,dp=C\,p^2F(p)(Q-T)^2\,dp,\qquad
\boxed{N'(p)=\sqrt C\,(Q-T)}.
$$
This <momentum-bin Kurie ordinate> is linear in <kinetic energy> and vanishes at its endpoint. If instead using counts per unit <kinetic energy>, $dN/dT\propto F(p)pE_e(Q-T)^2$ in $c=1$ units; converting with $dT/dp=p/E_e$ recovers the supplied momentum-bin definition. No extra factor of $E_e$ should be left inside that definition of $N'$.
The graph's horizontal coordinate is \b[$T/(m_ec^2)$], not $T$ in MeV. Its intercept is approximately $3.9$–$4.0$, so
$$
\boxed{Q=T_{\max}\simeq2.0\,\mathrm{MeV}}.
$$
For a ground-to-ground branch between bare <nuclei>, the emitted <electron>'s rest <energy> must also be supplied:
$$
\boxed{[M_{\rm nuc}(^{91}\mathrm Y)-M_{\rm nuc}(^{91}\mathrm{Zr})]c^2
\simeq Q+m_ec^2\simeq2.5\,\mathrm{MeV}}.
$$
For neutral atomic <masses> the <mass> difference instead equals approximately $Q\simeq2.0\,\mathrm{MeV}$, neglecting small electron-binding corrections. The graph supports this approximate precision, not a precise evaluated <isotope> <mass>. Interpreting a branch to an excited daughter would require adding its <excitation energy>; the ground-state comparison assumes the ground-state branch intended here.
The <shell-model prediction for yttrium-91 beta decay> follows from shell filling: the <proton> orbitals above 28 fill $2p_{3/2}$ with four <protons> and $1f_{5/2}$ with six, reaching 38. Yttrium has its 39th <proton> in $2p_{1/2}$, giving $J_i^\pi=1/2^-$. Its 52 <neutrons> include the closed 50-neutron core and a pair in $2d_{5/2}$, coupled to zero in the simple paired <ground state>. Zirconium has 40 paired <protons> and 51 <neutrons>, leaving an unpaired $2d_{5/2}$ <neutron>, so $J_f^\pi=5/2^+$. Therefore
$$
\boxed{|\Delta J|=2,\quad\pi_f=-\pi_i:\qquad\text{first-forbidden unique beta decay}}.
$$
The single-particle change is <neutron> $d_{5/2}$ to <proton> $p_{1/2}$; the <parity> change and rank two prevent an allowed Fermi or Gamow–Teller contribution.
\b[The straight plotted line is a constant-shape idealization.] For the shell-model transition just inferred, a leading unique-forbidden shape factor contains $C(T)\propto p_e^2+p_\nu^2$ in $c=1$ units, with Coulomb/finite-size refinements. Consequently the uncorrected quantity specified in the paper is generally proportional to $(Q-T)\sqrt{C(T)}$, not exactly linear. A consistent <shape-corrected Kurie plot> also divides its ordinate by $\sqrt{C(T)}$. The source's exact straight-line construction and its forbidden-transition inference cannot both be exact with a Coulomb-only $F$; the approximate graphical endpoint can still be read as requested. Allowed and unique-forbidden spectral shapes are experimentally distinguishable.<Shapes of Allowed and Unique First-Forbidden β-Ray Spectra: 114In, 42K, 86Rb, 90Sr, and 90Y by H. Daniel and others>[1964 abstract; separate allowed and unique-forbidden spectrum measurements]{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2013/paper-1-kurie-shapes.png]
{title=Original illustrative comparison of constant-shape and unique-first-forbidden Kurie ordinates with the same endpoint; no source data points reproduced}
Back to article page
