To obtain a rough estimate of the mean lime required for uranium-238 to alpha-decay, let u...

To obtain a rough estimate of the mean lime required for uranium-238 to alpha-decay, let us approximate the combined electrostatic and strong nuclear potential energies by a rectangular potential barrier half as high as the actual 35 MeV maximum potential energy. Alpha particles (mass 4 u) of 4.3 MeV kinetic energy are incident. Let us also assume that the barrier extends from the radius of the nucleus, 7.4 fm, to the point where the electrostatic potential drops to 4.3 MeV (i.e., the classically forbidden region). Because U ∞ 1/r, this point is 35/4.3 times the radius of the nucleus, the point at which U(r) is 35 MeV.

(a) Use these crude approximations, the method sug­gested in Section 6.3, and the wide-barrier approximation to obtain a value for the time it takes to decay.

(b) To gain some appreciation of the difficulties in a theoretical prediction, work the exercise ’‘backward.” Rather than assuming a value for U0, use the known value of the mean time to decay for ura­nium-238 and infer the corresponding value of U0. Retain all other assumptions.

(c) Comment on the sensitivity of the decay time to the height of the potential barrier.