Ratio of Half Lives (Homework 7, Physics 325)

2. What is the ratio of the half-lives $tau^{238}_{1/2} /tau^{ 228} _{1/2}$ for alpha decay of $_{92}U228$ and $_{92}U238$ using the equations in Griffiths, Eq. (8.25-8.29) and the energies in the plot in Fig. 8.6.

From the graph

$E(238)=4.32 MeV$  and $E(228)=6.88 MeV$

Assuming

$E=(m_α*v^2)/2$  we have $v^2 (238)=(2*4.32MeV)/(4*1.66*10^{-27} =2.08*10^14 (m/s)^2$   and $v^2 (228)=3.32*10^14 (m/s)^2$

$v(238)=1.44*10^7 m/s$  and $v(228)=1.82*10^7  m/s$

The exponential coefficient is

$γ=K_1*Z/√E-K_2*sqrt{Z*r_1}$    with $K_1=1.98 (√MeV)$     and $K_2=1.485 (1/√fm)$  
For r1 one has the values

$r_1 (228)=1.07*∛228=6.537 fm$ and $r_1 (238)=1.07*∛238=6.631 fm$

So that (Z=92)

$γ(228)=1.98*92/√6.88-1.485*sqrt{92*6.537}=33.03$

$γ(238)=1.98*92/√4.32-1.485*sqrt{92*6.631}=50.96$

Half-lifetime is

$τ=(2r_1)/v*exp⁡(2γ)$

$τ(228)=(2*6.537*10^{-15})/(1.82*10^7)*exp⁡(2*33.03)=3.51*10^7  seconds$

$τ(238)=(2*6.631*10^{-15})/(1.44*10^7 )*exp⁡(2*50.96)=1.69*10^{23}  seconds$

$τ(238)/τ(228) =4.81*10^{15}$