Review problem. Consider a nucleus at rest, which then...

Review problem. Consider a nucleus at rest, which then spontaneously splits into two fragments of masses $m_{1}$ and $m_{2}$ . Show that the fraction of the total kinetic energy carried by fragment $m_{1}$ is $$ \frac{K_{1}}{K_{\text { tot }}}=\frac{m_{2}}{m_{1}+m_{2}} $$ and the fraction carried by $m_{2}$ is $$ \frac{K_{2}}{K_{\mathrm{tot}}}=\frac{m_{1}}{m_{1}+m_{2}} $$ assuming relativistic corrections can be ignored. Note: If the parent nucleus was moving before the decay, the fission products still share the kinetic energy as shown as long as all velocities are measured in the center-of-mass frame of reference, in which the total momentum of the system is zero.

Review problem. Consider a nucleus at rest, which then spontaneously splits into two fragments of masses $m_{1}$ and $m_{2}$ . Show that the fraction of the total kinetic energy carried by fragment $m_{1}$ is
$$
\frac{K_{1}}{K_{\text { tot }}}=\frac{m_{2}}{m_{1}+m_{2}}
$$
and the fraction carried by $m_{2}$ is
$$
\frac{K_{2}}{K_{\mathrm{tot}}}=\frac{m_{1}}{m_{1}+m_{2}}
$$
assuming relativistic corrections can be ignored. Note: If the parent nucleus was moving before the decay, the fission products still share the kinetic energy as shown as long as all velocities are measured in the center-of-mass frame of reference, in which the total momentum of the system is zero.

Key Concepts

Kinetic Energy Distribution in Two-Body Decay

In two-body decays, the distribution of kinetic energy among the decay products is determined by the conservation laws of momentum and energy. The fractions of kinetic energy carried by each fragment are inversely proportional to their masses. This concept explains why, in the given scenario, the lighter fragment obtains a larger fraction of the total kinetic energy compared to the heavier one.

Center-of-Mass Frame

The center-of-mass frame is a reference frame in which the total momentum of the system is zero. Analyzing particle decays in this frame simplifies the mathematics because it allows one to treat the momenta of the products as equal and opposite. This simplification is vital to deriving straightforward expressions for how the kinetic energy is partitioned between decay products.

Conservation of Momentum

This principle states that in an isolated system, the total momentum remains constant. In the context of a decay process, where a nucleus splits into two fragments, the momenta of the fragments must be equal and opposite in the center-of-mass frame, ensuring that the net momentum is zero. This concept is crucial as it directly relates to how the kinetic energy is distributed between the fragments.

Conservation of Energy

The conservation of energy principle asserts that the total energy of an isolated system remains constant over time. In a decay process, the mass-energy of the original nucleus is redistributed into the kinetic energies of the daughter fragments (after accounting for binding energy changes). This is key to calculating the fractions of kinetic energy carried by each fragment.

Transcript

Please give Ace some feedback