Nuclear fission barriers and fragment distributions

In a liquid-drop approximation, elongating a nearly spherical nucleus increases its surface area but lowers its Coulomb energy. For a small dimensionless quadrupole deformation, the leading change is proportional to twice the spherical surface energy minus the spherical Coulomb energy. This motivates a fissility parameter comparing those two contributions. Below its liquid-drop instability value the sphere has positive curvature, but a lower-energy separated configuration can still exist beyond a finite barrier.

This local expansion does not determine the barrier maximum, scission configuration, or preferred mass split. Large deformations require more shape coordinates, and shell and pairing corrections can create multiple humps and metastable minima. A nucleus below a barrier can undergo spontaneous fission by collective tunneling, with a rate depending on the action and inertia along the chosen path. Treating a positive final Q as a rate prediction, or equating spontaneous fission with disappearance of local curvature, would conflate these distinct questions.

Rest-mass differences determine the total energy available after all specified final states are named. At scission, much of that energy can appear as fragment kinetic energy, while primary fragments may retain intrinsic excitation. Later neutron emission, photons, beta electrons, recoil, and neutrinos distribute the remaining release along a decay chain. A ground-state Q-value is therefore not identical to the kinetic energy measured for one fragment, nor necessarily to energy deposited locally by every eventual product.

For an ideal binary split in the parent rest frame, the primary fragment momenta are equal and opposite before subsequent emission. The lighter fragment then carries more translational kinetic energy. Neutron evaporation changes fragment masses, momenta, and measured energies, so post-neutron quantities cannot be inserted uncritically into a primary two-body formula. Conservation checks should identify whether fragment mass numbers and Q-values refer to the initial compound system, the primary split, or the later products.

A primary single-fragment mass yield counts how often each mass appears, usually normalized to two fragments per binary event. Its integrated yield is therefore two, or two hundred percent under a common reporting convention. A distribution normalized instead as a random single-fragment probability sums to one. State the convention before interpreting peak heights. Complementary masses are correlated within an event, so drawing two masses independently from a single-fragment curve generally violates mass conservation.

Shell effects can favor asymmetric primary splits at some low excitation energies, while increasing excitation can change the pattern; neither symmetric nor asymmetric splitting is universal for all nuclei and conditions. Charge polarization, odd–even effects, excitation sharing, and neutron evaporation add further structure. A measured post-emission mass curve blends these ingredients with instrumental response. Models must be compared with observables defined at the same stage, and a successful average mass yield does not by itself validate predicted pair correlations or neutron distributions.