---
title: The Fission Barrier and Fragment Energetics
module: Nuclear Fission
moduleNumber: 9
lessonNumber: 1
order: 901
summary: >
  Fission is the large-amplitude collective deformation of a heavy nucleus into two
  fragments. The liquid-drop model sets a barrier from the competition between rising
  surface energy and falling Coulomb energy under quadrupole deformation, with the
  fissility parameter Z²/A measuring how close a nucleus is to instability. Bohr-Wheeler
  theory separates spontaneous from neutron-induced fission, the fragment mass yield is
  double-humped and asymmetric, about 200 MeV is released per event, and shell corrections
  add a second minimum that produces fission isomers.
topics: [Nuclear Fission]
sources:
  - book: Krane
    ref: "Ch. 13 — Nuclear Fission; §13.1 Why Fission Occurs, §13.2 Characteristics of Fission, §13.3 Energetics of Fission, §13.4 Controlled Fission Reactions"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion; §6-4 Nuclear Fission"
draft: false
---

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.[^krane-why]

## 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
$\alpha_2$,

$$
R(\theta) = R_0\bigl[1 + \alpha_2 P_2(\cos\theta)\bigr],
$$

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
gives[^krane-drop]

$$
E_S(\alpha_2) = E_S^{0}\Bigl(1 + \tfrac{2}{5}\alpha_2^2 + \cdots\Bigr),
\qquad
E_C(\alpha_2) = E_C^{0}\Bigl(1 - \tfrac{1}{5}\alpha_2^2 + \cdots\Bigr),
$$

with $E_S^0$ and $E_C^0$ the spherical values. The surface energy costs $\tfrac{2}{5}$ per
unit $\alpha_2^2$; the Coulomb energy pays back $\tfrac{1}{5}$. The change in the total
energy is

$$
\Delta E(\alpha_2) = \bigl(\tfrac{2}{5}E_S^{0} - \tfrac{1}{5}E_C^{0}\bigr)\alpha_2^2
= \tfrac{1}{5}\bigl(2E_S^{0} - E_C^{0}\bigr)\alpha_2^2.
$$

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.

$$
% caption: A spherical drop distorts through a prolate spheroid to a necked configuration
% and finally scission; the quadrupole amplitude alpha grows left to right along the path.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % sphere
  \draw[very thick] (1.0,0) circle (0.55);
  \node[black!70, anchor=north] at (1.0,-0.85) {sphere};
  % prolate spheroid
  \draw[very thick] (3.0,0) ellipse (0.42 and 0.72);
  \node[black!70, anchor=north] at (3.0,-0.9) {spheroid};
  % necked
  \draw[very thick]
    (5.0,0.75) .. controls (5.55,0.55) and (5.55,0.28) .. (5.18,0.1)
    .. controls (4.95,0) and (4.95,0) .. (5.18,-0.1)
    .. controls (5.55,-0.28) and (5.55,-0.55) .. (5.0,-0.75)
    .. controls (4.45,-0.55) and (4.45,-0.28) .. (4.82,-0.1)
    .. controls (5.05,0) and (5.05,0) .. (4.82,0.1)
    .. controls (4.45,0.28) and (4.45,0.55) .. (5.0,0.75) -- cycle;
  \node[black!70, anchor=north] at (5.0,-0.9) {neck};
  % two fragments
  \draw[very thick] (6.7,0) circle (0.38);
  \draw[very thick] (7.7,0) circle (0.38);
  \node[black!70, anchor=north] at (7.2,-0.85) {scission};
  \draw[->, black] (1.7,0) -- (2.35,0);
  \draw[->, black] (3.6,0) -- (4.25,0);
  \draw[->, black] (5.75,0) -- (6.2,0);
\end{tikzpicture}
$$

## The fissility parameter

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

$$
2 a_s A^{2/3} = a_c \frac{Z^2}{A^{1/3}}
\quad\Longrightarrow\quad
\left(\frac{Z^2}{A}\right)_{\!\text{crit}} = \frac{2 a_s}{a_c}.
$$

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**

$$
x = \frac{E_C^0}{2 E_S^0} = \frac{a_c Z^2/A^{1/3}}{2 a_s A^{2/3}}
= \frac{Z^2/A}{(Z^2/A)_{\text{crit}}}.
$$

A nucleus with $x \geq 1$ 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.

> **Definition (Fissility).** The fissility parameter $x = (Z^2/A)/(Z^2/A)_{\text{crit}}$
> is the ratio of the disruptive Coulomb energy to twice the cohesive surface energy of the
> undeformed nucleus. The liquid-drop fission barrier vanishes at $x = 1$; for $x < 1$ the
> barrier height falls monotonically as $x$ rises.

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.

$$
% caption: Deformation energy along the fission path. A light nucleus (upper curve) has a
% tall barrier; a heavy nucleus with larger fissility (lower curve) has a low saddle before
% the energy plunges toward scission and the separated fragments.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {deformation};
  \draw[->, black] (0,-1.9) -- (0,3.6) node[above, black!70] {energy};
  % light nucleus: tall barrier
  \draw[black, thick, dashed]
    (0.3,0.05)
    .. controls (1.6,0.4) and (2.4,2.9) .. (3.3,3.0)
    .. controls (4.2,3.1) and (4.9,1.6) .. (5.6,0.4)
    .. controls (6.2,-0.6) and (7.0,-1.4) .. (7.8,-1.6);
  \node[black, anchor=west, font=\scriptsize] at (3.1,3.15) {lighter nucleus};
  % heavy nucleus: low saddle
  \draw[acc, very thick]
    (0.3,-0.05)
    .. controls (1.5,0.15) and (2.1,1.05) .. (2.7,1.15)
    .. controls (3.3,1.25) and (3.9,0.6) .. (4.6,-0.2)
    .. controls (5.4,-1.0) and (6.6,-1.55) .. (7.8,-1.7);
  \fill[acc] (2.7,1.15) circle (1.7pt);
  \node[acc, anchor=south, font=\scriptsize] at (2.7,1.25) {saddle};
  \node[acc, anchor=west, font=\scriptsize] at (0.4,-0.55) {heavier nucleus};
  % barrier height bracket
  \draw[black, dashed] (0,1.15) -- (2.7,1.15);
  \node[anchor=east, black, font=\scriptsize] at (-0.05,0.6) {$E_b$};
  \node[black, anchor=west, font=\scriptsize] at (6.2,-1.35) {scission};
\end{tikzpicture}
$$

## 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.

> **Theorem (Even-odd fissionability rule).** A heavy nucleus with an odd neutron number
> fissions with thermal neutrons, because capture produces an even-even compound nucleus
> whose neutron separation energy is raised by the pairing term above the fission barrier.
> A nucleus with an even neutron number requires a fast neutron, because capture yields an
> odd-$N$ compound nucleus with a lower separation energy that falls short of the barrier.

The contrast between the two dominant uranium isotopes is quantitative:

| Compound nucleus | Target | $S_n$ (MeV) | $E_b$ (MeV) | $S_n - E_b$ | Thermal fission |
| --- | --- | --- | --- | --- | --- |
| ${}^{236}\mathrm{U}^{\ast}$ | ${}^{235}\mathrm{U}$ (odd $N$) | $6.5$ | $6.2$ | $+0.3$ | yes |
| ${}^{239}\mathrm{U}^{\ast}$ | ${}^{238}\mathrm{U}$ (even $N$) | $4.8$ | $6.6$ | $-1.8$ | no |

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.[^krane-induced]

## 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.[^krane-mass]

$$
% caption: Fragment mass yield for thermal fission of uranium-235. Two asymmetric peaks
% near A = 95 and A = 140 rise three orders of magnitude above the symmetric valley; the
% dashed curve shows the single symmetric hump at high excitation energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {fragment mass number};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {yield (log scale)};
  % y ticks: log yield
  \foreach \y/\lab in {0.6/0.01,1.8/0.1,3.0/1,3.9/10} {\draw[black] (-0.07,\y) -- (0.07,\y); \node[anchor=east, black, font=\scriptsize] at (-0.05,\y) {\lab};}
  % x ticks
  \foreach \x/\lab in {1.4/80,3.4/110,5.4/140,7.2/165} \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab};
  % double-humped yield curve
  \draw[acc, very thick]
    (0.6,0.35)
    .. controls (1.6,0.9) and (2.1,3.6) .. (2.7,3.7)
    .. controls (3.1,3.75) and (3.4,1.3) .. (4.0,1.05)
    .. controls (4.6,1.3) and (4.9,3.75) .. (5.3,3.7)
    .. controls (5.9,3.6) and (6.4,0.9) .. (7.4,0.35);
  \node[black, anchor=south, font=\scriptsize] at (2.7,3.75) {light peak};
  \node[black, anchor=south, font=\scriptsize] at (5.3,3.75) {heavy peak};
  \node[black, anchor=south, font=\scriptsize] at (4.0,1.1) {symmetric valley};
  % high-excitation symmetric hump
  \draw[black, thick, dashed]
    (2.2,0.4) .. controls (3.4,2.0) and (3.4,2.0) .. (4.0,2.1)
    .. controls (4.6,2.0) and (4.6,2.0) .. (5.8,0.4);
  \node[black, anchor=west, font=\scriptsize] at (5.4,1.7) {high energy};
\end{tikzpicture}
$$

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 $d \approx r_0(A_1^{1/3} + A_2^{1/3})$,

$$
E_K \approx \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0\,d}
\approx \frac{(38)(54)(1.44\ \mathrm{MeV\,fm})}{6.4\ \mathrm{fm}} \approx 170\ \mathrm{MeV},
$$

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.[^krane-energetics]

| Component | Energy (MeV) | Timescale |
| --- | --- | --- |
| Fragment kinetic energy | $168$ | prompt |
| Prompt neutrons | $5$ | $\sim 10^{-14}\ \mathrm{s}$ |
| Prompt gamma rays | $7$ | $\sim 10^{-14}\ \mathrm{s}$ |
| Fragment beta decay | $8$ | seconds to years |
| Delayed gamma rays | $7$ | seconds to years |
| Antineutrinos | $12$ | (escape) |
| Total | $\approx 207$ | |

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.

$$
% caption: Energy released per uranium-235 fission. Fragment kinetic energy dominates the
% ledger; the remainder is split among prompt neutrons and gammas and the delayed radiation
% of the beta-decaying fragments, with the antineutrino share leaving the system.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {MeV};
  \foreach \y/\lab in {0/0,1.5/60,3.0/120,4.2/168} {\draw[black] (-0.07,\y) -- (0.07,\y); \node[anchor=east, black, font=\scriptsize] at (-0.05,\y) {\lab};}
  % bars
  \draw[acc, very thick, fill=acc!16] (0.7,0) rectangle (1.8,4.2);
  \node[black!70, anchor=north, align=center, font=\scriptsize] at (1.25,-0.1) {fragment\\KE};
  \draw[very thick] (2.2,0) rectangle (3.3,0.125);
  \node[black!70, anchor=north, align=center, font=\scriptsize] at (2.75,-0.1) {prompt\\neutrons};
  \draw[very thick] (3.7,0) rectangle (4.8,0.175);
  \node[black!70, anchor=north, align=center, font=\scriptsize] at (4.25,-0.1) {prompt\\gammas};
  \draw[black, very thick, fill=black!8] (5.2,0) rectangle (6.3,0.375);
  \node[black!70, anchor=north, align=center, font=\scriptsize] at (5.75,-0.1) {delayed\\beta gamma};
  \draw[black, very thick, dashed, fill=black!5] (6.7,0) rectangle (7.8,0.30);
  \node[black, anchor=north, align=center, font=\scriptsize] at (7.25,-0.1) {neutrinos\\(escape)};
\end{tikzpicture}
$$

## 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 $56\ \mathrm{s}$.

$$
% caption: Delayed-neutron emission. A neutron-rich fragment beta-decays to an excited
% daughter that lies above its neutron separation energy and promptly emits a neutron, so
% the neutron is released with the beta-decay half-life of the precursor.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % precursor level
  \draw[very thick] (0.2,3.0) -- (2.0,3.0);
  \node[black!70, anchor=east] at (0.15,3.0) {precursor};
  % beta decay arrow down to daughter excited state
  \draw[->, thick] (2.3,2.9) -- (3.6,2.15);
  \node[black, anchor=south west, font=\scriptsize] at (2.5,2.4) {beta decay};
  % daughter excited state above Sn
  \draw[black, very thick] (3.7,2.0) -- (5.5,2.0);
  \node[black!70, anchor=west] at (5.55,2.0) {excited daughter};
  % Sn line
  \draw[black, dashed] (3.7,1.35) -- (6.6,1.35);
  \node[black, anchor=west, font=\scriptsize] at (5.55,1.35) {$S_n$};
  % neutron emission arrow
  \draw[->, acc, thick] (4.6,1.9) -- (4.6,0.6);
  \node[acc, anchor=west, font=\scriptsize] at (4.7,1.2) {neutron out};
  % ground state of final nucleus
  \draw[black, very thick] (3.7,0.4) -- (5.5,0.4);
  \node[black!70, anchor=west] at (5.55,0.4) {ground state};
\end{tikzpicture}
$$

## 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.[^krane-isomer]

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.

$$
% caption: The double-humped fission barrier. The Strutinsky shell correction adds a second
% minimum to the smooth liquid-drop curve; a nucleus trapped in it is a superdeformed
% fission isomer bounded by an inner and an outer barrier.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {deformation};
  \draw[->, black] (0,-0.4) -- (0,3.6) node[above, black!70] {energy};
  % smooth liquid-drop barrier (dashed)
  \draw[black, thick, dashed]
    (0.4,0.3) .. controls (1.8,0.7) and (3.0,2.5) .. (4.2,2.55)
    .. controls (5.4,2.6) and (6.6,1.2) .. (7.9,-0.2);
  \node[black, anchor=west, font=\scriptsize] at (2.0,2.0) {liquid drop};
  % double-humped curve
  \draw[acc, very thick]
    (0.4,0.25)
    .. controls (1.3,0.55) and (1.7,2.55) .. (2.4,2.6)
    .. controls (2.9,2.63) and (3.2,1.15) .. (3.9,1.1)
    .. controls (4.6,1.05) and (4.9,2.7) .. (5.5,2.65)
    .. controls (6.2,2.6) and (6.9,1.0) .. (7.9,-0.25);
  \fill[black] (2.4,2.6) circle (1.6pt);
  \fill[black] (5.5,2.65) circle (1.6pt);
  \node[black, anchor=south, font=\scriptsize] at (2.4,2.7) {inner};
  \node[black, anchor=south, font=\scriptsize] at (5.5,2.75) {outer};
  % second minimum marker
  \draw[black, dashed] (3.9,0) -- (3.9,1.1);
  \fill[acc] (3.9,1.1) circle (1.6pt);
  \node[acc, anchor=north, align=center, font=\scriptsize] at (3.9,1.02) {isomer\\well};
  \node[black, anchor=north, font=\scriptsize] at (0.7,0.2) {ground};
\end{tikzpicture}
$$

> **Result.** Fission is controlled by the liquid-drop barrier set by the fissility
> $x = (Z^2/A)/50$, modulated by shell corrections into a double-humped shape. Whether a
> nucleus fissions with thermal neutrons hinges on the pairing swing in the neutron
> separation energy of the compound nucleus, the fragment yield is asymmetric because of
> the $N=82$, $Z=50$ closed shells, and each event releases about $200\ \mathrm{MeV}$,
> most of it as fragment kinetic energy, along with $\bar\nu \approx 2.4$ neutrons that
> make a chain reaction possible.

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](/nuclear-physics/fission/chain-reactions-reactor-physics).

[^krane-why]: **Krane**, _Introductory Nuclear Physics_, §13.1 (Why Fission Occurs). The
binding-energy argument, the surface-versus-Coulomb competition, and the smallness of the
barrier relative to the total energy release. Fission-fragment yields and neutron
multiplicities are compiled by the IAEA Nuclear Data Services,
[https://www-nds.iaea.org/](https://www-nds.iaea.org/), and the NNDC,
[https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).

[^krane-drop]: **Krane**, §13.3, and **Wong**, _Introductory Nuclear Physics_, §6-4. The
quadrupole expansion of the surface and Coulomb energies, $E_S = E_S^0(1 + \tfrac{2}{5}
\alpha_2^2)$ and $E_C = E_C^0(1 - \tfrac{1}{5}\alpha_2^2)$, and the critical condition
$Z^2/A = 2a_s/a_c$ are the Bohr-Wheeler liquid-drop analysis (1939).

[^krane-induced]: **Krane**, §13.1–13.2. The compound-nucleus excitation, the pairing swing
in $S_n$, and the thermal fissility of ${}^{233}\mathrm{U}$, ${}^{235}\mathrm{U}$, and
${}^{239}\mathrm{Pu}$ versus the fast-fission threshold of ${}^{238}\mathrm{U}$. Separation
energies from the AME evaluation via NNDC.

[^krane-mass]: **Krane**, §13.2 (Characteristics of Fission). The asymmetric double-humped
mass yield, the fixed heavy peak near $A = 140$, the shell stabilization at $N = 82$ and
$Z = 50$, and the transition to symmetric division at high excitation.

[^krane-energetics]: **Krane**, §13.3 (Energetics of Fission). The Coulomb estimate of
fragment kinetic energy and the full energy ledger (fragments, prompt neutrons and gammas,
delayed beta/gamma, antineutrinos) totaling about $207\ \mathrm{MeV}$ with $\sim 195\
\mathrm{MeV}$ recoverable.

[^krane-isomer]: **Krane**, §13.3, and **Wong**, §6-4. The Strutinsky shell correction, the
second minimum and superdeformed fission isomers such as ${}^{242m}\mathrm{Am}$, and the
intermediate-resonance structure in sub-barrier fission cross sections.