Uranium-238 is radioactive. A uranium nucleus (mass 238 u, charge 92 e ) at rest

Question

Uranium-238 is radioactive. A uranium nucleus (mass 238 u, charge 92 e) at rest decays by emitting an alpha particle (mass 4u, charge 2e; essentially a helium nucleus), leaving behind a smaller (daughter) nucleus. This process can by modeled in the following way: the original uranium nucleus is basically a 7.4fm radius sphere. Two protons and two neutrons “pinch off” from the rest to form an alpha particle that is initially at rest and situated 7.4 fm from the center of the daughter nucleus (the remaining nucleons). These two positively charged particles will then repel, “emitting” the alpha.

What is the mass (in u) and charge (in e) of the daughter nucleus? What is the name of this chemical element?

What is the total electrostatic potential energy of the alpha and daughter right after forming?

Assuming all this potential energy becomes kinetic energy of the particles, find the final speeds of both after they have moved far apart. Express your answer both in m/s and as a fraction of the speed of light, c. [Hint: this is a collision problem (an explosion). What is conserved?]

Explanation / Answer

The process of alpha decay in a nuclear reaction that can be written as:

Thus the -decay of 238U can be written:

The Q-value is positive (exothermic) for spontaneous alpha decay. The helium nucleus emerges with a substantial velocity and is fully ionized, and the atomic electrons on the daughter are disrupted by the sudden change but the whole process conserves electrical charge. Thus, we can rewrite the equation in terms of the masses of the neutral atoms:

and then calculate the Q-value because the net change in the atomic binding energies (~65.3 Z7/5 - 80 Z2/5 eV) is very small compared to the nuclear decay energy.

From the semi-empirical mass equation, the emission of an -particle lowers the Coulomb energy of the nucleus, which increases the stability of heavy nuclei while not affecting the overall binding energy per nucleon because the tightly bound -particle has approximately the same binding energy/nucleon as the original nucleus.

Two important features of alpha decay are that the energies of the alpha particles are known to generally increase with the atomic number of the parent but yet the kinetic energy of the emitted particle is less than that of the Coulomb barrier in the reverse reaction between the -particle and the daughter nucleus. In addition, all nuclei with mass numbers greater than A>>150 are thermodynamically unstable against alpha emission (Q is positive) but alpha emission is the dominant decay process only for the heaviest nuclei, A>210. The energies of the emitted -particles can range from 1.8 MeV (144Nd) to 11.6 MeV (212Pom) with the half-­life of 144Nd being 5x1029 times as long as that of 212Pom. Typical heavy element alpha decay energies are in the range from 4 to 9 MeV.

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