Beta Decay — Question 8

Question 8

A Kurie transformation linearizes an ideal allowed beta spectrum

An allowed spectrum is modeled as $dN/dT=CpE(Q-T)^2$, where $E=T+m_ec^2$ is total electron energy and p is momentum. Ignore Coulomb and recoil corrections. Define $K(T)=\sqrt{(dN/dT)/(pE)}$.

1. Derive the straight-line form of the Kurie plot.

2. Two measurements give $K(.200\,\mathrm{MeV})=8.00$ and $K(.600\,\mathrm{MeV})=4.00$ in consistent arbitrary units. Infer Q and predict K at .800 MeV.

Question 8 - Solution

A Kurie transformation linearizes an ideal allowed beta spectrum

(a) Method and calculation

$$K(T)=\sqrt{C(Q-T)^2}=\boxed{\sqrt C(Q-T)}\quad(0<T<Q).$$$$\boxed{\text{slope}=-\sqrt C,\quad K(Q)=0}.$$

(b) Interpretation and further calculation

$$\sqrt C=(8-4)/(.600-.200)=10.0\Rightarrow\boxed{Q=.200+8/10=1.00\,\mathrm{MeV}}.$$$$\boxed{K(.800)=10(1.00-.800)=2.00}.$$

Physical check

The horizontal intercept gives the ideal endpoint. Curvature can arise from omitted corrections, detector response or a superposition of transitions.