Chapter 20 — Nuclear Fission: Splitting the Atom

"The energy produced by the breaking down of the atom is a very poor kind of thing. Anyone who expects a source of power from the transformation of these atoms is talking moonshine." — Ernest Rutherford, 1933

He was wrong by six years.

Introduction

On December 17, 1938, Otto Hahn and Fritz Strassmann bombarded uranium with neutrons and found barium among the reaction products. Barium has $Z = 56$ — roughly half the atomic number of uranium ($Z = 92$). No known nuclear reaction could transmute uranium into barium. On December 21, Hahn wrote to his long-time collaborator Lise Meitner, who had fled Nazi Germany to Sweden months earlier: "Perhaps you can suggest some fantastic explanation." Over Christmas, Meitner and her nephew Otto Frisch worked out that explanation while cross-country skiing near Kungälv. The uranium nucleus was not chipping off small fragments — it was splitting roughly in half. Frisch, borrowing terminology from biology, named the process fission.

Within weeks, Meitner and Frisch published their explanation using the liquid drop model that Niels Bohr and John Archibald Wheeler would formalize later that year. The physics was already implicit in the semi-empirical mass formula (Chapter 4): heavy nuclei sit on the descending slope of the binding energy per nucleon curve, and splitting them into two medium-mass fragments releases energy because the fragments are more tightly bound per nucleon. What Bohr and Wheeler provided was the mechanism — the energetics of nuclear deformation that determines whether fission actually occurs and how fast.

This chapter develops the physics of fission from the liquid drop model through the chain reaction. We begin with the competition between surface energy and Coulomb energy that determines the fission barrier, derive the fissility parameter, and explain why some nuclei fission spontaneously while others require a neutron to trigger the process. We then examine what comes out: the peculiar asymmetric mass distribution of fission fragments, prompt and delayed neutrons, and the ~200 MeV energy release per fission event. Finally, we build the physics of the chain reaction — the multiplication factor, criticality, and the four-factor formula — and survey the modern landscape from Gen III+ reactors to Small Modular Reactors and the long-term challenge of nuclear waste.

Threshold Concept: Fission is not the shattering of a rigid object. It is the slow deformation and eventual rupture of a charged liquid drop. The same physics that gave us the surface and Coulomb terms of the SEMF (Chapter 4) governs whether a nucleus can fission and how large the barrier is. If you understood Chapter 4, you already have the essential tools.

📊 Spaced Review (Chapter 4): The SEMF binding energy is $B(Z,A) = a_V A - a_S A^{2/3} - a_C Z(Z-1)/A^{1/3} - a_{\text{sym}}(A-2Z)^2/A + \delta$. The surface term ($a_S A^{2/3}$) favors spherical shapes; the Coulomb term ($a_C Z(Z-1)/A^{1/3}$) favors elongated shapes that spread the charge. Fission is what happens when Coulomb wins.

📊 Spaced Review (Chapter 18): The compound nucleus model: when a neutron is absorbed, it forms a highly excited compound nucleus $A^* = A + 1$ with excitation energy equal to the neutron separation energy plus the kinetic energy of the incoming neutron. The compound nucleus then decays — and fission is one of its decay channels.

20.1 The Liquid Drop Model of Fission

Deformation and the Competition of Energies

Consider a nucleus in its ground state, approximately spherical with radius $R_0 = r_0 A^{1/3}$. Now imagine deforming it into a prolate (cigar-shaped) ellipsoid while keeping the volume constant — nuclear matter is nearly incompressible, so the density stays fixed. We parametrize the deformation by the quadrupole deformation parameter $\varepsilon$, such that the semi-axes of the ellipsoid are:

$$a = R_0(1 + \varepsilon), \qquad b = R_0(1 + \varepsilon)^{-1/2}$$

The volume is $V = \frac{4}{3}\pi a b^2 = \frac{4}{3}\pi R_0^3$, independent of $\varepsilon$. What happens to the energy as the nucleus deforms?

Surface energy. A sphere has the minimum surface area for a given volume. Any deformation increases the surface area and therefore increases the surface energy. For small $\varepsilon$, the surface area of the ellipsoid expands as:

$$S(\varepsilon) = 4\pi R_0^2 \left(1 + \frac{2}{5}\varepsilon^2 + \cdots\right)$$

Therefore the surface energy becomes:

$$E_S(\varepsilon) = a_S A^{2/3}\left(1 + \frac{2}{5}\varepsilon^2 + \cdots\right)$$

The surface energy increases with deformation — it acts as a restoring force that resists fission.

Coulomb energy. Spreading the proton charge over a more elongated shape decreases the average distance between charge elements, reducing the Coulomb energy. For the same ellipsoidal deformation:

$$E_C(\varepsilon) = a_C \frac{Z(Z-1)}{A^{1/3}}\left(1 - \frac{1}{5}\varepsilon^2 + \cdots\right)$$

The Coulomb energy decreases with deformation — it is the driving force for fission.

The Total Deformation Energy

The change in total energy upon deformation (to second order in $\varepsilon$) is:

$$\Delta E(\varepsilon) = E_S(\varepsilon) + E_C(\varepsilon) - E_S(0) - E_C(0)$$

$$\Delta E(\varepsilon) = \left(\frac{2}{5} a_S A^{2/3} - \frac{1}{5} a_C \frac{Z(Z-1)}{A^{1/3}}\right) \varepsilon^2$$

$$\boxed{\Delta E(\varepsilon) = \frac{1}{5}\left(2E_S^{(0)} - E_C^{(0)}\right)\varepsilon^2}$$

where $E_S^{(0)} = a_S A^{2/3}$ and $E_C^{(0)} = a_C Z(Z-1)/A^{1/3}$ are the surface and Coulomb energies of the undeformed nucleus.

The sign of the coefficient determines stability: - If $2E_S^{(0)} > E_C^{(0)}$: the coefficient is positive, and the spherical shape is a local minimum. The nucleus is stable against small deformations. Fission requires surmounting an energy barrier. - If $2E_S^{(0)} < E_C^{(0)}$: the coefficient is negative, and the spherical shape is a saddle point. Any small deformation lowers the energy, and the nucleus is unstable against fission — it fissions immediately.

The Fissility Parameter

We define the fissility parameter:

$$\boxed{x = \frac{E_C^{(0)}}{2E_S^{(0)}} = \frac{a_C Z(Z-1)}{2 a_S A^{1/3} \cdot A^{2/3}} = \frac{a_C}{2a_S}\frac{Z(Z-1)}{A} \approx \frac{a_C}{2a_S}\frac{Z^2}{A}}$$

where the last approximation holds for large $Z$. The critical condition for instability is $x = 1$, or equivalently:

$$\frac{Z^2}{A} \bigg|_{\text{crit}} = \frac{2a_S}{a_C} \approx \frac{2 \times 17.8}{0.714} \approx 49.8$$

Using the standard SEMF coefficients $a_S \approx 17.8$ MeV and $a_C \approx 0.714$ MeV, we find $(Z^2/A)_{\text{crit}} \approx 49$–$50$.

💡 Physical Insight: The fissility parameter $x$ measures how close a nucleus is to spontaneous instability. For $x < 1$, there is a fission barrier; for $x \geq 1$, there is none. No naturally occurring nucleus reaches $x = 1$, but heavy actinides come close: for $^{238}$U, $Z^2/A = 92^2/238 \approx 35.6$, giving $x \approx 0.71$. For $^{252}$Cf, $Z^2/A = 98^2/252 \approx 38.1$, giving $x \approx 0.76$.

20.2 The Fission Barrier

Beyond the Quadrupole: The Full Barrier Shape

The quadrupole analysis of Section 20.1 describes only the curvature of the potential energy surface near the spherical configuration. To find the actual fission barrier — the energy that must be supplied to drive the nucleus from its ground state to the "scission point" where it splits — we must follow the deformation to much larger values of $\varepsilon$.

The full deformation energy landscape, as calculated by Bohr and Wheeler in their landmark 1939 paper, looks qualitatively like this:

1. Near the ground state ($\varepsilon \approx 0$): the energy rises quadratically (for $x < 1$), reflecting the positive curvature we derived above.
2. At the saddle point ($\varepsilon = \varepsilon_{\text{saddle}}$): the energy reaches a maximum. This is the fission barrier $B_f$.
3. Beyond the saddle point: the nucleus is committed to fission. The two nascent fragments separate, and the Coulomb repulsion between them drives them apart, releasing kinetic energy.
4. At scission ($\varepsilon = \varepsilon_{\text{scission}}$): the neck connecting the two fragments breaks, and two separate fragments fly apart.

The barrier height $B_f$ depends on the fissility parameter $x$. Bohr and Wheeler showed that for a nucleus described by the liquid drop model alone:

$$B_f(x) \approx E_S^{(0)} \cdot f(x)$$

where $f(x)$ is a function that decreases from $f(0) = 1$ (maximum barrier for uncharged nuclei) to $f(1) = 0$ (no barrier at the critical fissility). A useful approximation for $0.7 \lesssim x < 1$ is:

$$B_f \approx 0.83 E_S^{(0)} (1 - x)^3 \text{ MeV}$$

For the actinides ($x \approx 0.7$–$0.76$):

📊 Numerical example: For $^{236}$U ($Z = 92$, $A = 236$):

$$E_S^{(0)} = a_S A^{2/3} = 17.8 \times 236^{2/3} = 17.8 \times 38.2 \approx 680 \text{ MeV}$$

$$x \approx \frac{0.714 \times 92^2}{2 \times 17.8 \times 236} \approx 0.72$$

$$B_f \approx 0.83 \times 680 \times (1 - 0.72)^3 \approx 0.83 \times 680 \times 0.022 \approx 12.4 \text{ MeV}$$

The experimental fission barrier for $^{236}$U is approximately 5.8 MeV. The liquid drop model overestimates the barrier because it ignores shell effects, which lower the barrier substantially for actinide nuclei. Including shell corrections (Strutinsky method, Chapter 6) brings theory into agreement with experiment.

⚠️ Common Misconception: Students sometimes think the fission barrier is the total energy released by fission (~200 MeV). It is not. The barrier is the activation energy — the relatively modest energy (~5–7 MeV for actinides) needed to push the nucleus over the saddle point. Once over, the enormous Coulomb repulsion between the fragments provides the ~200 MeV kinetic energy release.

Shell Corrections: The Strutinsky Method

The liquid drop model systematically overestimates fission barriers for actinide nuclei. The reason is that it ignores the quantum shell structure that becomes important at specific deformations. Veniamin Strutinsky (1967) developed a method to incorporate shell effects as a correction to the smooth liquid drop energy:

$$E_{\text{total}}(\varepsilon) = E_{\text{LD}}(\varepsilon) + \delta E_{\text{shell}}(\varepsilon)$$

where $E_{\text{LD}}$ is the liquid drop energy and $\delta E_{\text{shell}}$ is the shell correction — the difference between the actual quantum level density and the smoothed (Thomas-Fermi) level density at the given deformation. When the single-particle levels cluster (as they do near magic numbers), $\delta E_{\text{shell}}$ is negative (stabilizing). When levels spread apart, $\delta E_{\text{shell}}$ is positive (destabilizing).

For the ground states of actinide nuclei, the shell correction is moderately negative ($\delta E_{\text{shell}} \approx -2$ to $-4$ MeV), which deepens the ground-state minimum relative to the liquid drop alone. At the saddle-point deformation, the shell correction is typically positive or less negative, which reduces the effective barrier height. The net effect: shell corrections lower actinide fission barriers from the liquid drop prediction of ~10–15 MeV to the experimental values of ~5–7 MeV.

The Strutinsky method was a breakthrough: it showed that the smooth part of nuclear binding is well described by the liquid drop model, while the fluctuating part (shell effects) provides essential corrections for quantitative accuracy.

The Double-Humped Barrier

The liquid drop model predicts a single-humped barrier. However, experimental evidence — particularly from fission isomers, which are metastable states in the "second well" of the potential energy surface — reveals that the actinide fission barrier is double-humped. The origin is shell effects: as the nucleus deforms through the saddle region, it passes through a deformation where shell closures in the deformed potential create a local energy minimum (the second well) between two barrier peaks.

The physical picture is as follows. Starting from the ground state (approximately spherical or mildly deformed), the nucleus encounters the first barrier peak at moderate deformation. Beyond this, the single-particle energy levels rearrange, and a new set of shell closures — associated with a superdeformed shape (axis ratio approximately 2:1) — creates a secondary minimum. The nucleus must then surmount a second barrier before reaching the scission point.

The double-humped barrier has profound consequences: - Fission isomers (shape isomers): nuclei trapped in the second well have lifetimes of nanoseconds to milliseconds. These are nuclei with the same $Z$ and $N$ as the ground state but in a metastable superdeformed configuration. Over 30 fission isomers have been observed, primarily in the actinide region. - Subbarrier resonances: the tunneling probability through the double barrier shows resonance structure — narrow peaks in the fission cross section at energies corresponding to quasi-bound states in the second well. These "class-II" resonances were first observed in $^{240}$Pu and provide direct evidence for the second well. - The second barrier is typically the higher one for lighter actinides ($^{230}$Th, $^{232}$U) and the lower one for heavier actinides ($^{240}$Pu, $^{244}$Cm), reflecting the evolving shell structure with deformation.

For $^{240}$Pu, the two barrier heights are approximately $B_A \approx 6.05$ MeV and $B_B \approx 5.15$ MeV, with the second well about 2–3 MeV above the ground state. The fission isomer $^{240m}$Pu has a half-life of 3.7 ns, decaying primarily by fission through the lower outer barrier.

20.3 Spontaneous Fission

Tunneling Through the Barrier

If the fission barrier $B_f$ is nonzero but the nucleus is in its ground state (zero excitation energy), fission can still occur by quantum tunneling through the barrier — exactly the same physics that governs alpha decay (Chapter 15).

The spontaneous fission half-life depends exponentially on the barrier penetrability:

$$T_{1/2}^{\text{SF}} \propto \exp\left(\frac{2}{\hbar}\int_{r_1}^{r_2}\sqrt{2\mu[V(r) - E]}\,dr\right)$$

where the integral is over the classically forbidden region beneath the barrier, $\mu$ is the relevant reduced mass for the fission degree of freedom, and $V(r) - E$ is the barrier height above the ground state energy.

Because the barrier is wide and the relevant mass is large (the entire nuclear mass participates in the collective deformation), spontaneous fission half-lives are extremely sensitive to the barrier parameters. Small changes in $B_f$ produce enormous changes in half-life:

The trend is clear: as $Z^2/A$ increases (and the barrier decreases), the spontaneous fission half-life plummets. For the heaviest known elements ($Z > 110$), spontaneous fission is often the dominant decay mode with half-lives of milliseconds or less.

The systematic trend is approximately:

$$\log_{10} T_{1/2}^{\text{SF}} \approx a - b \cdot \frac{Z^2}{A}$$

where $a$ and $b$ are empirically fitted constants. For even-even actinides, $a \approx 120$ and $b \approx 2.8$ reproduce the general trend (the Swiatecki systematics), though individual half-lives can deviate by several orders of magnitude due to shell effects. The extreme sensitivity to $Z^2/A$ — a change of 3 units in $Z^2/A$ changes the half-life by roughly $10^{10}$ — is characteristic of tunneling phenomena.

💡 Physical Insight: Spontaneous fission sets the ultimate limit on the existence of superheavy elements. Without shell stabilization (the "island of stability"), no nucleus with $Z \gtrsim 104$ could survive long enough to be detected. The predicted island near $Z = 114$, $N = 184$ exists precisely because shell effects raise the fission barrier above the liquid drop prediction. The observation of $^{288}$Fl and $^{294}$Og with half-lives of milliseconds to seconds (compared to the microsecond timescale predicted by the liquid drop model alone) confirms that shell stabilization is real and significant in this region.

🔗 Connection to Chapter 15 (Alpha Decay): Both alpha decay and spontaneous fission involve quantum tunneling through a Coulomb barrier. The mathematical formalism (WKB approximation, Gamow factor) is identical. The key difference is the barrier shape: alpha decay involves tunneling of a compact $^4$He nucleus through a relatively narrow barrier, while spontaneous fission involves the collective motion of the entire nucleus through a broader, more complex barrier. The result is that spontaneous fission half-lives are generally much longer than alpha-decay half-lives for the same nucleus — the barrier for fission is wider and involves a larger effective mass.

20.4 Induced Fission and the Role of Pairing Energy

Adding Energy to Overcome the Barrier

If a nucleus does not fission spontaneously on a practical timescale, we can induce fission by supplying enough excitation energy to overcome (or at least significantly reduce) the fission barrier. The most common method is neutron capture: a neutron is absorbed by the target nucleus, forming an excited compound nucleus, and if the excitation energy exceeds the fission barrier, fission occurs.

The excitation energy of the compound nucleus after capturing a neutron of kinetic energy $E_n$ (in the center-of-mass frame) is:

$$E^* = S_n + E_n$$

where $S_n$ is the neutron separation energy of the compound nucleus — the binding energy gained by adding the last neutron. For thermal neutrons ($E_n \approx 0.025$ eV), the excitation energy is essentially just $S_n$.

Why $^{235}$U Fissions with Thermal Neutrons but $^{238}$U Does Not

This is one of the most important questions in nuclear physics, and the answer lies in the pairing term of the SEMF (Chapter 4).

Consider the two reactions:

$$n + {}^{235}\text{U} \to {}^{236}\text{U}^* \qquad (S_n = 6.55 \text{ MeV})$$

$$n + {}^{238}\text{U} \to {}^{239}\text{U}^* \qquad (S_n = 4.81 \text{ MeV})$$

The fission barriers are approximately: - $B_f(^{236}\text{U}) \approx 5.8$ MeV - $B_f(^{239}\text{U}) \approx 6.2$ MeV

For $^{235}$U + thermal $n$: $E^* = 6.55$ MeV $> B_f = 5.8$ MeV. Fission occurs.

For $^{238}$U + thermal $n$: $E^* = 4.81$ MeV $< B_f = 6.2$ MeV. Fission does not occur (at least not promptly — subbarrier fission is negligible at this energy).

The 1.74 MeV difference in neutron separation energies is the key. But why is $S_n$ so much larger for $^{236}$U than for $^{239}$U?

The answer is pairing. When $^{235}$U (odd-$N$, with $N = 143$) captures a neutron, the compound nucleus $^{236}$U has $N = 144$ — even-even. The last neutron pairs with the previously unpaired neutron, gaining the pairing energy $\delta \approx 12/\sqrt{A} \approx 0.78$ MeV (each neutron gets roughly half the pairing energy, but more precisely, the even-even nucleus benefits from a full pairing gap). This makes $S_n$ large.

When $^{238}$U (even-$N$, with $N = 146$) captures a neutron, the compound nucleus $^{239}$U has $N = 147$ — odd-$N$. The added neutron breaks a pair. There is no pairing energy bonus — in fact, the neutron must go into an unpaired state above the pairing gap. This makes $S_n$ smaller.

The general rule:

$$\boxed{\text{Odd-}N \text{ targets} + n \to \text{even-}N \text{ compound nucleus: large } S_n, \text{ fission with thermal } n}$$ $$\boxed{\text{Even-}N \text{ targets} + n \to \text{odd-}N \text{ compound nucleus: small } S_n, \text{ requires fast } n}$$

The "fissile" isotopes — those that fission with thermal neutrons — all have odd $N$: $^{233}$U ($N = 141$), $^{235}$U ($N = 143$), $^{239}$Pu ($N = 145$), $^{241}$Pu ($N = 147$). The "fissionable but not fissile" isotopes — those that require fast neutrons (typically $> 1$ MeV) — have even $N$: $^{238}$U ($N = 146$), $^{232}$Th ($N = 142$), $^{240}$Pu ($N = 146$).

⚠️ Terminology note: Fissile means "fissions with thermal neutrons." Fissionable means "can be made to fission" (with sufficiently energetic neutrons). All fissile nuclei are fissionable, but not all fissionable nuclei are fissile. $^{238}$U is fissionable but not fissile — it requires neutrons with $E_n \gtrsim 1$ MeV.

20.5 Fission Product Distributions

The Asymmetric Mass Distribution

If fission were simply the liquid drop splitting at the point of maximum symmetry, we would expect the two fragments to have roughly equal masses: $A_{\text{heavy}} \approx A_{\text{light}} \approx A/2$. Instead, for thermal-neutron fission of $^{235}$U, the mass distribution is strikingly asymmetric, with two peaks:

• Light fragment group: centered near $A_L \approx 95$ (Sr, Y, Zr, Mo region)
• Heavy fragment group: centered near $A_H \approx 140$ (Ba, La, Ce, Nd region)

The ratio is approximately $A_H / A_L \approx 1.4$, not $1.0$. Symmetric fission ($A_L = A_H = 118$) occurs but is suppressed by roughly two orders of magnitude relative to asymmetric fission.

This asymmetry was a major puzzle for decades. The liquid drop model predicts symmetric fission (the symmetric split maximizes the total kinetic energy release). The resolution lies in nuclear shell structure.

Shell Effects and the Heavy Fragment

The heavy fragment mass peak is remarkably stable: it sits near $A \approx 140$ regardless of the fissioning system. Whether we fission $^{233}$U, $^{235}$U, $^{239}$Pu, or $^{252}$Cf, the heavy fragment peak remains near $A \approx 132$–$140$. The light fragment peak shifts to accommodate the different total mass.

The explanation is the double shell closure at $Z = 50$ (magic proton number) and $N = 82$ (magic neutron number) in the heavy fragment. A fragment near $^{132}$Sn ($Z = 50$, $N = 82$) or slightly heavier gains substantial shell stabilization energy. The fissioning nucleus "prefers" to break in a way that places one fragment near this doubly-magic configuration, even though this means an asymmetric split.

📊 Numerical data for thermal fission of $^{235}$U:

The independent fission yield $Y(A)$ (probability of producing a fragment of mass $A$) shows the characteristic double-humped distribution. The peak yields are approximately 6%–7% per fragment, centered at: - Light peak: $A \approx 95$ with $\sigma \approx 6$ - Heavy peak: $A \approx 140$ with $\sigma \approx 6$ - Valley (symmetric region, $A \approx 118$): yield $\approx 0.01$%

The most probable fission products include $^{95}$Zr, $^{95}$Mo, $^{99}$Tc, $^{131}$I, $^{133}$Cs, $^{137}$Cs, $^{140}$Ba, and $^{144}$Ce.

Dependence on the Fissioning System and Excitation Energy

The asymmetry of the mass distribution is not universal — it depends on both the fissioning nucleus and the excitation energy:

Dependence on the parent nucleus. For thermal-neutron fission of different actinides, the heavy fragment peak is nearly stationary (anchored by the $Z = 50$, $N = 82$ shells), while the light fragment peak shifts:

The stability of the heavy peak is strong evidence for the shell-closure mechanism. As the total mass increases, the extra nucleons go predominantly to the light fragment.

Dependence on excitation energy. At higher excitation energies (fast neutrons, $E_n > 5$ MeV), the mass distribution becomes progressively more symmetric. The valley between the two peaks fills in, and at very high energies ($E^* > 40$ MeV), the distribution approaches the symmetric Gaussian predicted by the liquid drop model. This occurs because the shell correction energy becomes less important relative to the thermal excitation energy — high temperature "washes out" shell effects, just as in atomic physics (electron shells become less distinct in hot, dense plasmas).

The transition from asymmetric to symmetric fission is a beautiful demonstration of the interplay between collective (liquid drop) and single-particle (shell) degrees of freedom. At low excitation, shell structure dominates and fission is asymmetric. At high excitation, the liquid drop dominates and fission becomes symmetric. The crossover occurs at $E^* \approx 20$–$40$ MeV for actinides.

Charge Distribution and Unchanged Charge Distribution (UCD)

In the initial scission, the charge-to-mass ratio of each fragment is approximately equal to that of the fissioning nucleus — this is the unchanged charge distribution (UCD) assumption:

$$\frac{Z_f}{A_f} \approx \frac{Z_{\text{parent}}}{A_{\text{parent}}}$$

The fragments are therefore born extremely neutron-rich (because the parent actinide has $N/Z \approx 1.55$), far from stability. This neutron excess drives the prompt neutron emission and the extensive beta-decay chains that follow.

20.6 Fission Neutrons and Gammas

Prompt Neutrons

The fission fragments are born with a large neutron excess. Some of this excess is shed immediately (within $\sim 10^{-14}$ s of scission) as prompt neutrons — neutrons evaporated from the highly excited fragments.

For thermal fission of $^{235}$U, the average number of prompt neutrons per fission is:

$$\bar{\nu}_p = 2.43$$

This is an average — the actual number in any single fission event follows a distribution (roughly Gaussian, centered at $\bar{\nu}_p$ with a width of about 1.1 neutrons). The range is from 0 to about 6, with 2 and 3 being the most probable values.

The prompt neutron energy spectrum follows the Watt fission spectrum, well approximated by:

$$N(E) \propto \exp(-E/a)\sinh(\sqrt{bE})$$

where $a \approx 0.988$ MeV and $b \approx 2.249$ MeV$^{-1}$ for $^{235}$U thermal fission. The average prompt neutron energy is $\langle E_n \rangle \approx 2.0$ MeV, and the most probable energy is about 0.73 MeV. The spectrum extends from near-zero to about 10 MeV, with a tail beyond that.

These are fast neutrons — their average energy of 2 MeV corresponds to a speed of approximately $2 \times 10^7$ m/s. To induce fission in $^{235}$U efficiently, they must be slowed down ("moderated") to thermal energies ($E \approx 0.025$ eV at 300 K, speed $\sim 2200$ m/s). The slowing-down process requires approximately $\ln(2 \times 10^6 / 0.025) / \bar{\xi} \approx 18 / 1.0 \approx 18$ collisions with hydrogen (in a light-water reactor) or $\sim 115$ collisions with carbon (in a graphite reactor). This is because each collision with hydrogen reduces the neutron's kinetic energy by a factor of $\sim 2$ on average ($\bar{\xi} = 1$), while carbon reduces it by a factor of $\sim 1.16$ ($\bar{\xi} = 0.158$).

Prompt Gammas

In addition to neutrons, the excited fission fragments emit prompt gamma rays. On average, about 7–8 prompt gammas are emitted per fission, carrying a total energy of approximately 7 MeV. The gamma-ray spectrum is approximately exponential, with most photons in the 0.5–1 MeV range but a tail extending to several MeV.

Delayed Neutrons

After prompt neutron emission, the fission fragments undergo beta-decay chains toward stability. In a small fraction of cases (roughly 1.6% for $^{235}$U fission), a beta decay populates an excited state in the daughter nucleus that lies above the neutron separation energy. This state then emits a neutron rather than a gamma ray. These are delayed neutrons — "delayed" because they are emitted on the timescale of the parent beta decay (seconds to minutes), not on the nuclear timescale.

The delayed neutron fraction $\beta$ is defined as:

$$\beta = \frac{\bar{\nu}_d}{\bar{\nu}} = \frac{\bar{\nu}_d}{\bar{\nu}_p + \bar{\nu}_d}$$

where $\bar{\nu}_d$ is the average number of delayed neutrons per fission. For $^{235}$U: $\bar{\nu}_d \approx 0.0158$, so $\beta \approx 0.0065$ (0.65%).

Delayed neutrons are grouped into six conventional groups based on their precursor half-lives:

The weighted mean half-life is approximately 13 seconds, giving an effective delayed neutron time constant of about 13 seconds. As we will see in Section 20.8, this ~13-second timescale is the reason nuclear reactors can be controlled by humans and mechanical systems rather than requiring response times of microseconds.

⚠️ This point cannot be overemphasized: Without delayed neutrons, controlled nuclear chain reactions would be essentially impossible with any practical control system. The prompt neutron generation time in a thermal reactor is $\ell \sim 10^{-4}$ s. If the reactor power changed on this timescale, no mechanical control rod could respond fast enough. Delayed neutrons stretch the effective generation time from $10^{-4}$ s to $\sim 0.1$ s, making control feasible. This is a gift of nuclear physics to nuclear engineering.

20.7 Energy Release in Fission

The 200 MeV Budget

The total energy released per fission of $^{235}$U is approximately 200 MeV. This is an enormous amount of energy for a single nuclear event — roughly $10^8$ times the energy released in a chemical combustion reaction per molecule.

The energy is distributed among several components:

The neutrino energy escapes the reactor entirely (neutrinos interact too weakly to be absorbed). Therefore, the recoverable energy per fission is approximately 196 MeV.

Derivation from the Binding Energy Curve

We can estimate the fission energy release directly from the binding energy per nucleon curve (Chapter 4):

For $^{236}$U: $B/A \approx 7.59$ MeV, so $B(^{236}\text{U}) \approx 7.59 \times 236 = 1791$ MeV.

For two typical fragments, $^{95}$Mo and $^{139}$La (with 2 neutrons released): - $B(^{95}\text{Mo}) \approx 8.64 \times 95 = 821$ MeV - $B(^{139}\text{La}) \approx 8.38 \times 139 = 1165$ MeV - Free neutrons: $B = 0$

The total binding energy of the products is $821 + 1165 = 1986$ MeV, while the parent has $B = 1791$ MeV. The energy released is:

$$Q = B_{\text{products}} - B_{\text{parent}} = 1986 - 1791 = 195 \text{ MeV}$$

in good agreement with the detailed accounting.

💡 Physical Insight: The energy comes from the increase in binding energy per nucleon as we move from the heavy actinide region ($B/A \approx 7.6$ MeV) to the medium-mass region ($B/A \approx 8.5$ MeV). The difference of $\sim 0.9$ MeV per nucleon, multiplied by $\sim 236$ nucleons, gives $\sim 200$ MeV per fission event. This is the same binding energy curve from Chapter 4 — fission exploits the descending slope on the heavy side.

Energy per Unit Mass

One kilogram of $^{235}$U contains $N = 6.022 \times 10^{23} / 0.235 \approx 2.56 \times 10^{24}$ atoms. If each fission releases 196 MeV of recoverable energy:

$$E = 2.56 \times 10^{24} \times 196 \times 1.602 \times 10^{-13} \text{ J} = 8.04 \times 10^{13} \text{ J} \approx 80 \text{ TJ}$$

This is equivalent to about 19,000 tonnes of TNT, or roughly 2,700 tonnes of coal. The energy density of nuclear fuel is approximately two million times that of chemical fuels.

20.8 The Chain Reaction

Concept and the Multiplication Factor

The emission of $\bar{\nu} \approx 2.43$ neutrons per $^{235}$U fission creates the possibility of a self-sustaining chain reaction: each fission produces neutrons that can induce further fissions. Whether the reaction sustains itself, grows, or dies out depends on how many of those neutrons actually cause new fission events.

We define the effective neutron multiplication factor $k_{\text{eff}}$:

$$k_{\text{eff}} = \frac{\text{number of fission neutrons in generation } (n+1)}{\text{number of fission neutrons in generation } n}$$

Three regimes:

For a reactor operating at steady power, $k_{\text{eff}} = 1$ exactly. The power is then determined by the neutron population, which can be adjusted by briefly making the reactor slightly supercritical (to increase power) or slightly subcritical (to decrease power).

The Four-Factor Formula

For an infinite, homogeneous reactor (no neutron leakage), the multiplication factor is given by the four-factor formula:

$$\boxed{k_\infty = \eta \cdot f \cdot p \cdot \varepsilon}$$

Each factor represents a step in the neutron life cycle:

1. The reproduction factor $\eta$ ("eta"):

$$\eta = \nu \cdot \frac{\sigma_f}{\sigma_a}$$

where $\nu = \bar{\nu}$ is the average number of neutrons per fission, $\sigma_f$ is the microscopic fission cross section, and $\sigma_a = \sigma_f + \sigma_\gamma$ is the total absorption cross section of the fuel (fission + radiative capture). For $^{235}$U with thermal neutrons: $\sigma_f = 584$ b, $\sigma_\gamma = 99$ b, so $\sigma_a = 683$ b, and:

$$\eta = 2.43 \times \frac{584}{683} = 2.08$$

This means each thermal neutron absorbed in $^{235}$U produces, on average, 2.08 new neutrons.

2. The thermal utilization factor $f$:

$$f = \frac{\Sigma_a^{\text{fuel}}}{\Sigma_a^{\text{fuel}} + \Sigma_a^{\text{moderator}} + \Sigma_a^{\text{other}}}$$

This is the probability that a thermal neutron is absorbed in the fuel rather than in the moderator, structural materials, or other absorbers. In a well-designed reactor, $f \approx 0.71$–$0.95$, depending on the fuel-to-moderator ratio.

3. The resonance escape probability $p$:

$$p = \exp\left(-\frac{N_{\text{fuel}} I_{\text{res}}}{\bar{\xi} \Sigma_s}\right)$$

This is the probability that a neutron slowing down from fission energies to thermal energies avoids capture in the resonance absorption peaks of $^{238}$U (and other resonance absorbers). The parameter $I_{\text{res}}$ is the resonance integral, $\bar{\xi}$ is the average logarithmic energy decrement per collision with the moderator, and $\Sigma_s$ is the macroscopic scattering cross section. For typical light-water reactors, $p \approx 0.75$–$0.90$.

4. The fast fission factor $\varepsilon$:

This accounts for fissions caused by fast neutrons (above the $^{238}$U fission threshold, $\sim 1$ MeV) before the neutrons slow down to thermal energies. In most thermal reactors, $\varepsilon \approx 1.02$–$1.08$ — a small but non-negligible enhancement.

📊 Numerical example: A typical pressurized water reactor (PWR) with low-enriched uranium fuel:

The infinite multiplication factor $k_\infty = 1.48$ is considerably larger than 1. The actual effective multiplication factor is reduced by neutron leakage from the finite reactor volume:

$$k_{\text{eff}} = k_\infty \cdot P_{\text{NL}}$$

where $P_{\text{NL}}$ is the non-leakage probability (typically 0.94–0.97 for large power reactors). More precisely, $P_{\text{NL}}$ is the product of the fast non-leakage probability $P_{\text{FNL}}$ (probability that a fast neutron does not leak out while slowing down) and the thermal non-leakage probability $P_{\text{TNL}}$ (probability that a thermal neutron does not leak out before being absorbed). The full six-factor formula is therefore:

$$k_{\text{eff}} = \eta \cdot f \cdot p \cdot \varepsilon \cdot P_{\text{FNL}} \cdot P_{\text{TNL}}$$

For a large reactor, both leakage terms are close to 1, and the four-factor formula is a good approximation. For small reactors (like research reactors or SMRs), leakage is more significant and must be carefully accounted for.

The excess reactivity $k_{\text{eff}} - 1$ is controlled by absorber materials (control rods, soluble boron) and is needed to compensate for fuel depletion and fission product poisoning over the reactor operating cycle. The most important fission product poison is $^{135}$Xe, which has a thermal neutron absorption cross section of approximately $2.65 \times 10^6$ b — the largest of any known nuclide. After a reactor shutdown, $^{135}$Xe builds up from the beta decay of $^{135}$I (half-life 6.6 hours) faster than it is removed by neutron capture, temporarily increasing the negative reactivity. This "xenon poisoning" can prevent reactor restart for 24–48 hours after shutdown and was a contributing factor in the Chernobyl accident.

Generation Time, Reactivity, and Reactor Kinetics

The reactivity $\rho$ is defined as:

$$\rho = \frac{k_{\text{eff}} - 1}{k_{\text{eff}}}$$

For a critical reactor, $\rho = 0$. For a supercritical reactor, $\rho > 0$.

The neutron population $n(t)$ in a reactor changes according to the point reactor kinetics equations. In the simplest case (neglecting delayed neutrons temporarily), the prompt neutron generation time is:

$$\ell = \frac{1}{v \Sigma_a k_{\text{eff}}}$$

where $v$ is the average neutron speed and $\Sigma_a$ is the macroscopic absorption cross section. For a thermal reactor, $\ell \sim 10^{-4}$–$10^{-3}$ s. The neutron population then evolves as:

$$n(t) = n(0) \exp\left(\frac{\rho}{\ell}t\right)$$

If $\rho = 0.003$ (a small positive reactivity) and $\ell = 10^{-4}$ s, the $e$-folding time would be $\ell/\rho = 0.033$ s — the power would increase by a factor of $e^{30} \approx 10^{13}$ in one second. This is uncontrollable.

Prompt Critical vs. Delayed Critical

The crucial distinction:

Delayed critical ($0 < \rho < \beta$): The reactor is supercritical, but only when delayed neutrons are counted. The effective generation time is not $\ell$ but rather a weighted average that includes the delayed neutron precursor lifetimes:

$$\ell_{\text{eff}} \approx \frac{\beta - \rho}{\lambda_{\text{eff}}} + \frac{\ell}{1 - \rho}$$

where $\lambda_{\text{eff}} \approx 0.08$ s$^{-1}$ is an effective delayed neutron decay constant. For small positive reactivity ($\rho \ll \beta$), the effective $e$-folding time is on the order of seconds to tens of seconds — readily controllable.

Prompt critical ($\rho = \beta$): The reactor is critical on prompt neutrons alone. The delayed neutrons become irrelevant for criticality. The power changes on the prompt generation timescale $\ell \sim 10^{-4}$ s — far too fast for mechanical control. A reactor must never reach prompt critical during normal operations.

The reactivity is measured in units of the delayed neutron fraction $\beta$. One dollar (\$) of reactivity equals $\rho = \beta$, and one **cent** equals $\rho = \beta/100$. A power reactor typically operates with reactivity perturbations of a few cents — safely below the prompt critical threshold.

$$\boxed{\text{The delayed neutron fraction } \beta \approx 0.0065 \text{ is the safety margin that makes controlled fission possible.}}$$

20.9 The Manhattan Project: The Physics Decisions

The story of the atomic bomb is primarily a story of politics, ethics, and human tragedy. Here we focus narrowly on the physics decisions that translated laboratory fission into a weapon, because they illustrate the chapter's core concepts under extreme conditions.

The critical mass problem. A nuclear weapon requires a supercritical assembly that remains supercritical for long enough to release significant energy before it blows itself apart. The critical mass depends on the fuel, geometry, neutron reflectors, and density. For a bare sphere of $^{235}$U at normal density, the critical mass is approximately 52 kg. With a natural uranium reflector, this drops to about 15 kg.

Enrichment. Natural uranium is 99.3% $^{238}$U and only 0.7% $^{235}$U. For a weapon, highly enriched uranium (HEU, >90% $^{235}$U) is needed. The Manhattan Project developed two enrichment methods: electromagnetic separation (calutrons at Y-12, Oak Ridge) and gaseous diffusion (K-25, Oak Ridge). Both exploit the tiny mass difference between $^{235}$UF$_6$ and $^{238}$UF$_6$ molecules.

Plutonium and the first chain reaction. Enrico Fermi's Chicago Pile-1 (CP-1), which achieved the first self-sustaining chain reaction on December 2, 1942, was a milestone in human history — the moment when nuclear energy passed from theory to reality. The pile was constructed under the stands of Stagg Field at the University of Chicago, using 385 tonnes of graphite as moderator and approximately 45 tonnes of natural uranium (metal and oxide) as fuel. It operated at a maximum power of about 200 watts (deliberately limited — there was no radiation shielding beyond the graphite itself). The pile demonstrated both that a chain reaction was feasible and that $^{239}$Pu could be produced by neutron capture in $^{238}$U:

$$^{238}\text{U} + n \to {}^{239}\text{U} \xrightarrow{\beta^-} {}^{239}\text{Np} \xrightarrow{\beta^-} {}^{239}\text{Pu}$$

Plutonium-239 is fissile ($\eta = 2.11$ for thermal neutrons) and can be chemically separated from uranium — a far simpler process than isotope enrichment.

The gun design vs. implosion. The uranium bomb (Little Boy) used a gun-type design: a subcritical mass of $^{235}$U was fired into another subcritical mass to create a supercritical assembly. This design is simple but inefficient. It could not be used for plutonium because reactor-produced Pu contains a significant fraction of $^{240}$Pu, which has a high spontaneous fission rate. The spontaneous fission neutrons would initiate the chain reaction prematurely, before the assembly reached optimal supercriticality ("predetonation"), resulting in a fizzle. The solution was the implosion design (Fat Man): a subcritical sphere of Pu was compressed to supercritical density by converging detonation waves from a shell of conventional explosives. Compression increases the density (hence the macroscopic cross sections), reduces the critical mass, and allows the assembly to become supercritical faster than spontaneous fission neutrons can cause predetonation.

🔗 Connection to Section 20.3: The spontaneous fission rate of $^{240}$Pu ($T_{1/2}^{\text{SF}} = 1.14 \times 10^{11}$ yr, emitting $\sim 10^6$ neutrons per second per kg) is the physics that forced the development of the implosion design — one of the most technically challenging engineering achievements of the 20th century.

20.10 Modern Fission Technology

Current Reactor Fleet

As of 2024, approximately 440 nuclear power reactors operate worldwide, providing about 10% of global electricity. The dominant reactor types are:

• Pressurized Water Reactors (PWR): ~300 units. Water serves as both moderator and coolant, kept liquid at ~155 bar. Fuel is low-enriched UO$_2$ (3–5% $^{235}$U).
• Boiling Water Reactors (BWR): ~60 units. Water boils directly in the core at ~75 bar.
• CANDU (Pressurized Heavy Water Reactor): ~30 units. Heavy water (D$_2$O) moderator allows use of natural uranium fuel.
• RBMK (Graphite-moderated, Water-cooled): ~10 units (Russia). Graphite moderator with direct-cycle light water coolant — the type involved in the 1986 Chernobyl accident.

Generation III+ and Small Modular Reactors

Gen III+ reactors (AP1000, EPR, VVER-1200, APR1400) incorporate passive safety systems that rely on natural circulation, gravity, and compressed gas rather than active pumps requiring electrical power. The AP1000, for example, has a gravity-driven emergency coolant injection system and a passive containment cooling system.

Small Modular Reactors (SMRs) represent a potential paradigm shift. Key designs include:

• NuScale VOYGR (77 MWe per module): An integral PWR with all primary components inside a single vessel. The reactor operates inside a below-grade water pool; in a loss-of-coolant accident, the pool provides passive cooling. NuScale received NRC design certification in 2023.

• GE-Hitachi BWRX-300 (300 MWe): A simplified BWR design with approximately 60% fewer components and 50% less building volume than conventional BWRs. Under construction at the Darlington site in Ontario.

• Kairos Power Hermes (35 MWth): A fluoride-salt-cooled, pebble-bed reactor using TRISO fuel — tristructural-isotropic coated fuel particles with multiple containment layers. Construction permit granted 2023.

SMRs offer potential advantages: factory fabrication (reducing construction time and cost), flexible siting, load-following capability, and inherent safety features. The physics is the same — thermal fission with $k_{\text{eff}} = 1$ — but the engineering is more compact and modular.

The Nuclear Waste Challenge

Fission produces radioactive waste that must be managed for long periods. The waste falls into two categories with fundamentally different timescales:

Fission products (e.g., $^{137}$Cs, $^{90}$Sr, $^{99}$Tc, $^{129}$I): these are medium-mass nuclei near the peaks of the fission yield curve. The short-lived fission products ($T_{1/2} < 30$ years) dominate the radioactivity for the first few hundred years. After $\sim 300$ years (about 10 half-lives of $^{137}$Cs and $^{90}$Sr), the fission product activity drops to levels comparable to the original uranium ore.

Transuranic actinides (Pu, Am, Cm, Np): these are produced by successive neutron captures in $^{238}$U and subsequent beta decays. They have half-lives of thousands to millions of years — $^{239}$Pu ($T_{1/2} = 24{,}110$ yr), $^{237}$Np ($T_{1/2} = 2.14 \times 10^6$ yr), $^{241}$Am ($T_{1/2} = 432$ yr). These dominate the waste radiotoxicity after the first few hundred years and are the primary reason waste repositories must isolate material for geological timescales.

Strategies for waste management:

1. Deep geological disposal: Burial in stable geological formations at depths of 500–1000 m. Finland's Onkalo repository (in crystalline bedrock) is the world's first, scheduled to begin accepting spent fuel in the 2020s. Sweden's Forsmark site is approved. The U.S. Yucca Mountain project remains in political limbo.

2. Transmutation: Converting long-lived actinides into shorter-lived or stable nuclides via neutron-induced fission or (n,$\gamma$) reactions in fast reactors or accelerator-driven systems. If the transuranic actinides could be fully fissioned, the remaining waste would be predominantly fission products, requiring isolation for only $\sim 300$ years instead of $> 100{,}000$ years.

3. Advanced fuel cycles: Reprocessing spent fuel to extract plutonium and uranium for recycling (as practiced in France's La Hague facility), and closing the fuel cycle with fast breeder reactors that can consume transuranics while producing energy.

⚖️ The double-edged sword: Nuclear fission produces no greenhouse gases during operation and has the highest energy density of any developed technology. It also produces radioactive waste and carries the risk (however small) of severe accidents. The physics is agnostic — it is the same whether the application is power generation, medical isotopes, or weapons. As Enrico Fermi reportedly said after CP-1: "The Italian navigator has just landed in the New World." What humanity builds in that new world is not a physics question.

20.11 Connecting the Physics

The Binding Energy Curve, Revisited

Everything in this chapter flows from the binding energy curve of Chapter 4:

1. Why fission releases energy: $B/A$ increases from $\sim 7.6$ MeV (actinides) to $\sim 8.5$ MeV (medium-mass fragments). The difference, multiplied by $A$, gives $\sim 200$ MeV.

2. Why heavy nuclei are fissionable: The Coulomb term in the SEMF grows as $Z^2/A^{1/3}$, eventually overwhelming the surface term. The fissility parameter $x = E_C/2E_S$ approaches 1 for heavy nuclei.

3. Why fission products are neutron-rich: The valley of stability curves toward $N > Z$ for heavy nuclei. When a heavy nucleus ($N/Z \approx 1.55$) splits into medium-mass fragments, the fragments retain the parent's neutron-to-proton ratio but find themselves far from their own stability valley ($N/Z \approx 1.3$ for $A \sim 100$).

4. Why prompt neutrons are emitted: The neutron-rich fragments can lower their energy by evaporating neutrons. The number of prompt neutrons is determined by the neutron excess and the fragment excitation energies.

5. Why the chain reaction works: The $\bar{\nu} \approx 2.43$ prompt neutrons per fission, combined with delayed neutrons, provide enough neutrons to sustain the chain reaction while leaving a controllable margin.

Looking Forward

Chapter 21 will turn to the other great application of the binding energy curve: nuclear fusion, which exploits the ascending slope on the light side ($B/A$ increasing from $\sim 1$ MeV for deuterium to $\sim 7$ MeV for helium). Where fission splits heavy nuclei to reach the peak of the curve from the right, fusion combines light nuclei to reach it from the left. Together, fission and fusion bracket the entire landscape of nuclear energy.

🔗 Connection to Chapter 22 (Nuclear Astrophysics): In stellar environments, both fission and fusion play roles. Fusion powers stars; fission (specifically, the spontaneous fission of heavy $r$-process nuclei) terminates the rapid neutron capture process and determines the upper boundary of the chart of nuclides. The fission barrier heights and fragment distributions calculated in this chapter are essential inputs to $r$-process nucleosynthesis models.

Chapter 20 Notation Reference

Fission is the physics of instability harnessed — or, depending on the application, unleashed. The liquid drop's failure to hold itself together against its own Coulomb repulsion powers reactors, produces medical isotopes, and created the weapon that reshaped geopolitics. The same five SEMF terms from Chapter 4 — volume, surface, Coulomb, asymmetry, pairing — determine the barrier, the yields, and the energy release. Nuclear physics rarely offers a more complete circle from fundamental theory to world-altering technology.