Nuclear Fission

Step 1: Parameterize the Deformation

Consider an axially symmetric quadrupole deformation of a sphere of radius $R_0$. The nuclear surface is described by:

where $\alpha_2$ is the quadrupole deformation parameter and $P_2(\cos\theta) = (3\cos^2\theta - 1)/2$is the Legendre polynomial. Volume conservation to second order requires:

Step 2: Surface and Coulomb Energies Under Deformation

The surface area of the deformed shape, expanded to second order in $\alpha_2$:

Therefore the surface energy increases:

The Coulomb energy of a uniformly charged deformed body decreases:

Step 3: Deformation Energy and Fissility

The total change in energy upon deformation is:

The nucleus is unstable against deformation when $\Delta E < 0$:

Defining the fissility parameter:

The critical value is $x = 1$, corresponding to:

For $x < 1$, a fission barrier exists (height depends on $x$). For $x \geq 1$, no barrier exists and the nucleus instantly fissions.

Barrier Height in the Liquid Drop Model

For a nucleus with fissility parameter $x < 1$, the fission barrier height in the liquid drop approximation is approximately:

For specific nuclei:

Double-Humped Fission Barrier

The liquid drop model predicts a single barrier, but shell effects create a double-humped barrier structure for actinide nuclei. At intermediate deformations ($\alpha_2 \approx 0.3\text{-}0.6$), shell effects produce a second minimum:

The structure of the double-humped barrier:

• - First minimum: Ground state (spherical or slightly deformed)
• - First barrier (A): Height $\sim 6$ MeV above ground state
• - Second minimum: Shape isomeric state, typically 2-3 MeV above ground state
• - Second barrier (B): Height $\sim 5$ MeV above ground state
• - Scission point: Nucleus divides into two fragments

The second minimum gives rise to fission isomers -- metastable states with lifetimes of nanoseconds to milliseconds that decay preferentially by fission rather than gamma emission back to the ground state.

Asymmetric Mass Division

Thermal fission of actinides produces an asymmetric mass distribution with two peaks. For $^{235}$U(n,f), the light fragment peaks at $A_L \approx 95$ and the heavy fragment at $A_H \approx 140$:

The asymmetry is driven by shell effects in the fragments, particularly the proximity of the heavy fragment to the doubly-magic $^{132}$Sn (Z=50, N=82). As the excitation energy increases, the mass distribution becomes more symmetric.

Kinetic Energy Distribution

The total kinetic energy (TKE) of the fragments is primarily Coulomb repulsion at the scission point:

where $D$ is the center-to-center distance at scission, typically$D \approx r_0(A_1^{1/3} + A_2^{1/3}) + 2$ fm. For $^{235}$U fission:

The Viola-Seaborg systematics give an empirical formula:$\langle\text{TKE}\rangle = 0.1189\,Z^2/A^{1/3} + 7.3$ MeV, which works well across the actinide region.

Prompt Neutrons

Fission fragments are born highly excited and neutron-rich. They de-excite by emitting prompt neutrons (within $\sim 10^{-14}$ s of scission):

The prompt neutron energy spectrum follows the Watt distribution:

with $a \approx 0.988$ MeV and $b \approx 2.249$ MeV$^{-1}$ for $^{235}$U. The average prompt neutron energy is $\langle E \rangle \approx 2$ MeV.

Delayed Neutrons

A small fraction ($\beta_{\rm eff} \approx 0.0065$ for $^{235}$U) of fission neutrons are emitted following beta decay of certain fission products (delayed neutron precursors). These are crucial for reactor control:

The delayed neutron fraction is grouped into six time groups with half-lives ranging from 0.2 to 55 seconds. The effective multiplication factor including delayed neutrons is:

Reactor control is possible only because the delayed neutron fraction provides a time scale of seconds (rather than microseconds) for power changes. Without delayed neutrons, the reactor period for a 0.1% excess reactivity would be $T \approx \ell/\delta k \sim 10^{-4}/10^{-3} = 0.1$ s (too fast to control). With delayed neutrons, $T \approx \beta/(\lambda\delta k) \sim 80$ s.

SF Half-Lives

The SF half-life depends exponentially on the barrier height and width. The action integral through the fission barrier is:

where $B(\alpha)$ is the collective inertia (mass parameter) and$V(\alpha)$ is the potential energy along the fission path. SF half-lives span an enormous range:

Ternary Particle Spectrum

The most common ternary particle is the alpha particle (~90% of ternary events), emitted roughly perpendicular to the fission axis:

• - Alpha particles: ~0.2% of all fissions, average energy ~16 MeV, emitted at ~82$^\circ$ to the fission axis
• - Tritons ($^3$H): ~0.01% of fissions, average energy ~8.5 MeV
• - Other light ions: $^6$He, $^8$He, $^{6-9}$Li, $^{7-11}$Be at much lower rates