The edge of a nucleus can be roughly modeled as a square potential barrier. An alpha particle in an unstable nucleus can be modeled as a particle with a specific energy, bouncing back and forth between these square potential barriers. Each time the alpha particle comes into contact with a potential barrier, there is some probability T that it will tunnel out. Since the tunneling probability is small, it may be approximated by T = Ge-2kl
where G = 16 E/U0(1 - E/U0) and k = √2m(Uo - E)/h
and where E is the total energy of the alpha particle, U0 is the potential energy of the barrier, m is the mass of the alpha particle, and h is Planck's constant divided by 2 pi. This is a complicated expression, but the important part for this problem is simply that T is a constant if we assume that the barrier and alpha particle's energies are both fixed.
Consider a nucleus of radius r and an alpha particle with kinetic energy E (i. e., let the potential energy within the nucleus be zero) and mass m.
a. Assuming that the alpha particle moves along a diameter of the nucleus and that it moves at low enough speed that relativistic effects are negligible, what is the time r between successive encounters between each edge of the nucleus and the alpha particle?
b. In a given period of time Delta t, what is the probability P(Delta t) that the nucleus has decayed (i. e., the probability that the alpha particle has actually tunneled out)?
c. If you have a large sample containing N unstable nuclei, how many - Delta N would be expected to decay during a period of time Delta t? Use P to stand for P(Delta t) that you found in Part B. Note that you are solving for the negative change in N, because we want to know how much the number N has decreased. The amount of decrease - Delta N is a positive quantity. Express your answer in terms of P and N.