PX156 - D7 - alpha decay of nuclei

geiger-nuttall law

\ln λ = - C_{1} \frac{Z - 2}{\sqrt{E}} + C_{2}

$λ =$ decay constant
$Z =$ atomic number
$E =$ energy of the emitted $α$ particle
$C_{1}, C_{2} =$ constants

the alpha particle is in the potential well of the nucleus
a classical particle must have an energy greater than the barrier to escape, but a quantum particle doesn't have to
classically, alpha decay cannot be explained
alpha decay happens due to quantum tunnelling
$N (t) =$ number of nuclei remaining at time, $t :$ $$\frac{dN}{dt}= -\lambda N$$
$τ = \frac{1}{λ} =$ life time

\begin{aligned} N (t_{\frac{1}{2}}) & = \frac{1}{2} N (0) = N (0) \exp (- λ t_{\frac{1}{2}}) \\ ∴ t_{\frac{1}{2}} & = \ln (\frac{2}{λ}) = τ \ln (2) \end{aligned}

an $α$ particle: $2 p^{+}, 2 n^{0}$
atomic number of nucleus $: Z$
charges $: Q_{1} = 2^{+}, Q_{2} = (Z - 2)^{+}$
potential energy:

V (r) = \frac{2 (Z - 2) e^{2}}{4 π ϵ_{0} r}

if the particle escapes:

E_{K} = V (r) = \frac{2 (Z - 2) e^{2}}{4 π ϵ_{0} d}

where, $d =$ width of barrier

\begin{matrix} d = \frac{(Z - 2) e^{2}}{2 π ϵ_{0} E_{K}} \\ T \sim \exp (- \frac{(Z - 2) e^{2} \sqrt{2 m (V - E_{K})}}{π ϵ_{0} E_{K} ℏ}) \end{matrix}

for $V >> E_{K} :$

T \sim \exp (- \frac{A (Z - 2)}{E_{K}})

where, $A =$ collection of constants

in reality, there are a range of kinetic energies, so to account for that, an integral must be taken:

\begin{aligned} T & \sim \exp (- \frac{2 \sqrt{2 m}}{ℏ} \int_{R}^{r^{'}} \sqrt{\frac{2 (Z - 2) e^{2}}{r} - E_{K}} d r) \\ = \exp (- C_{1} \frac{Z - 2}{\sqrt{E_{K}}}) \end{aligned}

$λ =$ decay rate = number of attempts at the barrier per unit time $\times T :$

\begin{aligned} λ & = \frac{v}{2 R} T \\ \ln λ & = - C_{1} \frac{Z - 2}{\sqrt{E}} + C_{2} \end{aligned}

where, $v =$ velocity if $α$ particle inside

note: $λ$ as well as $25$ orders of magnitude, while $E_{K}$ changes very little ( $4 - 9 M e V$ )