Question

Calculation of half-life for alpha emission using time-independent
Schrodinger Equation using the following following information:
Radionuclide: 241-Am (Z=95); Ea = 5.49 MeV; Measured Half-Life~432y

Follow the steps involved and show your work for each subset question, not the
final answer:
(a) Evaluate the well radius [=separation distance (r) between the center of
the alpha particle as it abuts the recoil nucleus];
(b) Evaluate the coulomb barrier potential energy (U) for the well;
(c) Estimate the separation distance (r*) from the center of the potential well
where the coulomb potential equals the energy of the alpha particle;
(d) Assuming a square shaped single barrier of height “U”, evaluate the
tunneling probability by solving for the transmission coefficient and use of
the associated separation (=r*-r);
(e) Calculate the frequency with which the alpha particle strikes the well
boundary to try to get out of the well;
(f) Calculate the half-life and compare it with the known half-life for alpha
emission from 238U;
(g) Use the spatially-averaged effective approximation for the height of the
barrier by integrating the variation of “U” with distance from “r” to “r*” and
re-calculate the half-life and compare it with the known half-life for alpha
emission from 238U;
(h) Approximate the hyperbolic shaped barrier variation outside of the well
by breaking them up into 5 progressively reduced height square shaped
barriers, each with a width = (r*-r)/5 and calculate the tunneling
probabilities associated with each segment;
(i) Re-calculate for the half-life combining the probabilities from each of the 5
bins, and compare the value with the known half-life.

Answer 1