Beta Decay — Question 7

Question 7

The same phase-space model predicts count fractions and endpoint sensitivity

Use the model $d\Gamma/dT=C_0T^2(Q-T)^2$, with the same constant $C_0$ for two otherwise identical transitions. Coulomb and electron-mass effects are neglected.

1. Find how total decay rate scales with Q and the half-life ratio if Q doubles.

2. For a normalized spectrum, calculate the fraction of electrons with $T<Q/4$.

Question 7 - Solution

The same phase-space model predicts count fractions and endpoint sensitivity

(a) Method and calculation

$$\Gamma=C_0\int_0^QT^2(Q-T)^2dT=\boxed{C_0Q^5/30}.$$$$Q\to2Q:\quad\boxed{\Gamma\prime/\Gamma=32,\quad T_{1/2}\prime/T_{1/2}=1/32}.$$

(b) Interpretation and further calculation

$$F(x)=30\int_0^xu^2(1-u)^2du=10x^3-15x^4+6x^5.$$$$\boxed{F(1/4)=.103515625\simeq10.35\%}.$$

Physical check

The fifth-power rule here assumes unchanged nuclear matrix element and the stated limiting spectrum. Nuclear selection rules can outweigh a simple Q comparison.