Question

Estimate the half-life for the alpha decay of Uranium. The energy of the alpha particle for Uranium is 6 eV and the diameter of the nucleus is 9.5 fm

Answer 1

given, energy of decay of uranium, dE = 6 eV
diameter of nucleus, d = 9.5 fm
let half life of alpha decay be t'
then

from uncertianity principle

dp*dx >= h/4*pi
where dp is uncertianity in momentum
dx is uncertianity in position

now we can write dp = m*dv
and m*dv = m*dx/dt
hence

m*dx^2/dt >= h/4*pi
now, mc^2 = dE
hence
m = dE/c^2
hence

dE(dx/c)^2*4*pi/h >= dt
dt <= (6 eV)(1.6*10^-19C)(9.5*10^-15/3*10^8)^2*4*pi/6.63*10^-34
dt <= 1.8246*10^-29 s