The Fission Barrier and Fragment Energetics

Fission is the division of a heavy nucleus into two fragments of comparable mass, with the release of several neutrons and about $200\mathrm{MeV}$. The binding energy per nucleon peaks near $A=56$ at $8.8\mathrm{MeV}$ and falls to $7.6\mathrm{MeV}$ for the actinides, so splitting a uranium nucleus into two mid-mass fragments recovers roughly $0.9\mathrm{MeV}$ per nucleon. Energy release alone does not make fission happen: the nucleus must first deform through a sequence of shapes whose energy rises before it falls, and the height of that barrier decides whether a nucleus fissions spontaneously in microseconds or survives for the age of the universe.¹

Deformation energy in the liquid-drop model

Model the nucleus as an incompressible charged drop. Two terms of the semi-empirical mass formula respond to a change of shape at fixed volume: the surface energy $E_S=a_s{A}^{2/3}$, which a sphere minimizes, and the Coulomb energy $E_C=a_c{Z}^2{A}^{-1/3}$, which any elongation reduces by spreading the charge apart. Their competition governs stability against fission.

Parametrize a small axially symmetric distortion of the surface by a quadrupole amplitude ${\alpha }_2$,

where $P_2$ is the Legendre polynomial and $R_0$ is adjusted to conserve volume to second order. Expanding the surface area and the electrostatic self-energy of the deformed drop gives²

with $E_S^0$ and $E_C^0$ the spherical values. The surface energy costs $\frac{2}{5}$ per unit ${\alpha }_2^2$; the Coulomb energy pays back $\frac{1}{5}$. The change in the total energy is

The sign of the coefficient decides everything. If $2E_S^0>E_C^0$ the sphere sits at a local minimum and small deformations cost energy; the nucleus is stable against infinitesimal distortion and can fission only by climbing over a barrier. If $2E_S^0<E_C^0$ the coefficient is negative, the sphere is unstable, and the drop flies apart with no barrier at all.

The fissility parameter

Insert the mass-formula coefficients. The instability condition $2E_S^0=E_C^0$ becomes

With $a_s\approx 17.8\mathrm{MeV}$ and $a_c\approx 0.71\mathrm{MeV}$, the critical ratio is $(Z^2/A{)}_{\text{crit}}\approx 50$. Define the dimensionless fissility parameter

A nucleus with $x\ge 1$ has no barrier and cannot exist; $x<1$ nuclei have a barrier that shrinks as $x$ approaches unity. Uranium-235 has $Z^2/A=92^2/235=36.0$, so $x\approx 0.72$; the heaviest actinides push toward $x\approx 0.8$, where the barrier is only a few MeV. This is why fission is a phenomenon of the heaviest nuclei: the Coulomb energy grows as $Z^2$ while the stabilizing surface energy grows only as $A^{2/3}$, and somewhere near the actinides the drop can no longer hold itself together against a large deformation.

The quadratic $\Delta E\propto (1-x){\alpha }_2^2$ only describes the initial rise. Following the deformation to large ${\alpha }_2$ requires higher multipoles, and the full liquid-drop calculation of Bohr and Wheeler gives a barrier that first rises to a saddle point and then falls steeply to scission, where the neck breaks. For actinides the barrier height $E_b$ is $5$–$6\mathrm{MeV}$, small compared with the $\sim 200\mathrm{MeV}$ eventually liberated but large enough to make spontaneous fission extraordinarily slow.

Spontaneous versus induced fission

A nucleus in its ground state lies at the bottom of the deformation well, below the saddle by the barrier height $E_b$. Two routes carry it over.

• Spontaneous fission proceeds by quantum tunneling through the barrier, exactly as in alpha decay but with a much heavier and more collective object penetrating a much thicker barrier. The tunneling probability is set by an action integral over the deformation coordinate, and the resulting partial half-lives are enormous and steeply dependent on $x$. For ${}^{238}\mathrm{U}$ the spontaneous-fission half-life is $\sim 10^{16}\mathrm{yr}$, many orders longer than its alpha half-life; for ${}^{252}\mathrm{Cf}$ ($x$ larger) it drops to $85\mathrm{yr}$, and spontaneous fission becomes a practical neutron source.
• Induced fission adds energy to the compound nucleus so that it is created at or above the saddle. Capturing a neutron on ${}^{235}\mathrm{U}$ forms ${}^{236}{\mathrm{U}}^{\ast }$ with an excitation equal to the neutron separation energy, $S_n=6.5\mathrm{MeV}$, plus the neutron's kinetic energy. Since $S_n$ already exceeds the barrier $E_b\approx 6.2\mathrm{MeV}$, even a zero-energy (thermal) neutron drives the nucleus over the top.

The decisive quantity is the difference $S_n-E_b$. The neutron separation energy is larger when the captured neutron pairs with an odd neutron to make an even-even compound nucleus. This pairing swing explains the sharpest fact of reactor physics.

The contrast between the two dominant uranium isotopes is quantitative:

Uranium-238 fissions only when the incident neutron supplies more than about $1.8\mathrm{MeV}$, so it is a fast-fission material and a fertile capture target, not a thermal fuel. The same rule makes ${}^{233}\mathrm{U}$ and ${}^{239}\mathrm{Pu}$, both with odd neutron number in the target, thermally fissile.³

The fragment mass distribution

Fission does not divide the nucleus into two equal halves. The yield $Y(A)$, the percentage of fissions producing a fragment of mass number $A$, is strongly asymmetric for thermal-neutron fission of the actinides, with two peaks near $A\approx 95$ and $A\approx 140$ and a deep valley at symmetric division. The heavy peak stays fixed near $A\approx 140$ across the actinides while the light peak shifts with the mass of the fissioning system, evidence that the heavy fragment is anchored by nuclear structure. The valley is attributed to the stabilizing influence of the closed shells at $N=82$ and $Z=50$, which favor a heavy fragment near ${}^{132}\mathrm{Sn}$. As the excitation energy rises the two peaks fill in, and above a few tens of MeV the distribution becomes single- humped and symmetric, confirming that the asymmetry is a low-energy shell effect and not a property of the liquid drop.⁴

Because the fragments carry the same neutron-to-proton ratio as the parent, which is far higher than the stable ratio at their smaller mass, they are born grossly neutron rich. They shed neutrons promptly and then beta-decay in chains toward stability, which is the origin of both the prompt fission neutrons and the long-lived radioactive waste.

The energy ledger

The total energy released, $Q_f\approx 200\mathrm{MeV}$, is the difference between the binding of the parent and the fragments. It appears in several forms, most of it as the kinetic energy of the two fragments driven apart by their mutual Coulomb repulsion at scission. Estimate that repulsion by placing two point charges $Z_{1}e$ and $Z_{2}e$ at the scission separation $d\approx r_0(A_1^{1/3}+A_2^{1/3})$,

for a representative split near ${}^{95}\mathrm{Sr}+{}^{140}\mathrm{Xe}$. The Coulomb estimate lands close to the measured fragment kinetic energy of about $168\mathrm{MeV}$, confirming that most of the released energy is electrostatic. The rest is distributed among prompt neutrons, prompt gamma rays, and the beta, gamma, and antineutrino emission of the decaying fragments.⁵

The antineutrinos escape the reactor entirely, so the recoverable energy is about $195\mathrm{MeV}$ per fission; delayed emission from the fragments contributes the decay heat that must still be removed after a reactor shuts down.

Prompt and delayed neutrons

Each fission releases on average $\bar{\nu }\approx 2.4$ neutrons for thermal fission of ${}^{235}\mathrm{U}$, rising to $\bar{\nu }\approx 2.9$ for ${}^{239}\mathrm{Pu}$. The great majority are prompt, boiled off the fully accelerated fragments within about $10^{-14}\mathrm{s}$ with a Maxwellian spectrum peaked near $0.7\mathrm{MeV}$ and a mean energy of about $2\mathrm{MeV}$. A small fraction, the delayed neutrons, appear seconds to minutes later. They are emitted not directly in fission but by highly neutron-rich fragments whose beta decay populates a daughter state above its neutron separation energy, allowing prompt neutron emission from that state. The neutron therefore appears with the half-life of the beta-decaying precursor.

The canonical precursor is ${}^{87}\mathrm{Br}$, which beta-decays with a $55\mathrm{s}$ half-life to a ${}^{87}\mathrm{Kr}$ state above the ${}^{86}\mathrm{Kr}+n$ threshold. The delayed-neutron yield is small, a fraction $\beta \approx 0.0065$ of all neutrons for ${}^{235}\mathrm{U}$, but this handful of slow neutrons is what makes a reactor controllable, the subject of the next lesson. Delayed neutrons are conventionally sorted into six groups by precursor half-life, from $0.2\mathrm{s}$ to $56\mathrm{s}$.

Fission isomers and the double-humped barrier

The single-humped liquid-drop barrier is incomplete. Adding the shell correction of Strutinsky, an oscillating term that tracks the bunching of single-particle levels as the shape changes, modulates the smooth drop energy and produces a second minimum in the deformation-energy curve at a large, elongated deformation with an axis ratio near $2:1$. A nucleus caught in this second well is a fission isomer: a superdeformed state that can decay only by tunneling forward through the outer barrier to scission or backward through the inner barrier to the normal ground state. Fission isomers such as ${}^{242m}\mathrm{Am}$ have spontaneous-fission half-lives in the millisecond to nanosecond range, shorter than their ground states by more than twenty orders of magnitude, because they start their tunneling already high on the deformation path.⁶

The double-humped barrier also explains resonant structure in sub-barrier fission cross sections: states in the second well appear as sharp intermediate resonances when the excitation energy matches a level in that well, a direct spectroscopic signature of the second minimum predicted by the shell correction.

The $2.4$ neutrons per fission, of which a small delayed fraction lags by seconds, are the raw material for a self-sustaining chain reaction. Controlling that chain, through the neutron economy of a reactor core, is the subject of chain reactions and reactor physics.