Chapter 17 — Nuclear Reaction Fundamentals: Kinematics, Q-Values, and Cross Sections

"All science is either physics or stamp collecting." — Ernest Rutherford (attributed)

Rutherford, who said this, was of course the master of nuclear reactions — even if the term had not yet been coined when he scattered alpha particles off gold foil in 1911. The study of what happens when nuclei collide, rearrange, fuse, or break apart is the experimental engine of nuclear physics. Everything we know about nuclear structure, nuclear forces, and nuclear astrophysics ultimately rests on our ability to do things to nuclei and measure what comes out.

This chapter lays the quantitative foundation for all of Part IV. We will develop the language (reaction notation), the conservation laws, the kinematics (Q-values, thresholds, frame transformations), and the central observable quantity — the cross section — that governs every nuclear reaction from elastic scattering to fission. We will derive the Rutherford cross section in full, introduce the partial wave formalism that connects quantum mechanics to reaction observables, and develop the optical model that treats the nucleus as a cloudy crystal ball.

Fast Track: If you are comfortable with center-of-mass kinematics from classical mechanics, skim Sections 17.1--17.4 and begin in earnest at Section 17.5 (cross sections). The essential new material is the Rutherford derivation (Section 17.6), partial waves (Section 17.7), and the optical model (Section 17.8).

Deep Dive: The Rutherford derivation in Section 17.6 is carried out in full detail. If you have seen it before (Chapter 1 presented a sketch), the new element here is the rigorous connection to the partial wave formalism and the transition from classical to quantum scattering.

17.1 What Is a Nuclear Reaction?

17.1.1 Reaction Notation

A nuclear reaction involves a projectile $b$ striking a target $a$ (at rest in the laboratory) to produce ejectile $c$ and residual nucleus $d$:

$$a + b \to c + d$$

The compact notation is:

$$a(b, c)d$$

where $a$ is the target, $b$ is the projectile, $c$ is the light product (detected), and $d$ is the heavy product (often the residual nucleus). For example:

• ${}^{12}\text{C}(d, p){}^{13}\text{C}$ — deuteron stripping on carbon-12, producing a proton and carbon-13
• ${}^{197}\text{Au}(n, \gamma){}^{198}\text{Au}$ — neutron capture on gold, producing a gamma ray and gold-198
• ${}^{14}\text{N}(\alpha, p){}^{17}\text{O}$ — the reaction Rutherford used in 1919 to achieve the first artificial transmutation of an element

The notation extends naturally to reactions with more than two products: $a(b, cd)e$ for three-body final states, though two-body kinematics will be our primary focus.

17.1.2 Entrance and Exit Channels

The entrance channel specifies the projectile-target pair and their quantum numbers (kinetic energy, angular momentum, spin orientation). The exit channel specifies the products. A given entrance channel may have many possible exit channels. For example, bombarding ${}^{12}\text{C}$ with protons at 20 MeV can produce:

Which channels are open depends on the beam energy and conservation laws.

17.1.3 Classification of Nuclear Reactions

Nuclear reactions are classified by their mechanism and timescale:

1. Elastic scattering — $a(b, b)a$: The same particles emerge, with the same internal quantum numbers, but redirected in angle. No kinetic energy is converted to internal excitation ($Q = 0$). Elastic scattering is always the dominant reaction channel.

2. Inelastic scattering — $a(b, b')a^*$: The projectile loses kinetic energy to excite the target (or vice versa). The prime notation indicates that the outgoing projectile has a different energy. The Q-value equals the negative of the excitation energy: $Q = -E^*$.

3. Transfer reactions — One or more nucleons are transferred between projectile and target. Stripping reactions (e.g., $(d, p)$, $(d, n)$) transfer nucleons from the projectile to the target; pickup reactions (e.g., $(p, d)$, $(d, {}^{3}\text{He})$) transfer nucleons from the target to the projectile. These are the primary tools for studying single-particle structure (Chapter 19).

4. Charge-exchange reactions — $(p, n)$, $(n, p)$, $(\pi^+, \pi^0)$: A proton-neutron conversion occurs without net nucleon transfer. These probe the isospin structure of nuclei.

5. Capture reactions — $a(b, \gamma)c$: The projectile is absorbed by the target and the excess energy is radiated as one or more gamma rays. Radiative capture of neutrons is crucial in reactor physics and nucleosynthesis.

6. Compound nucleus reactions — The projectile and target fuse into an excited intermediate state that lives long enough ($\sim 10^{-18}$--$10^{-15}\,\text{s}$) to "forget" how it was formed. The subsequent decay is independent of the entrance channel. This is the subject of Chapter 18.

7. Direct reactions — The projectile interacts with individual nucleons at the nuclear surface in a single step, without forming a compound nucleus. Timescale: $\sim 10^{-22}\,\text{s}$ (the transit time of the projectile across the nucleus). This is the subject of Chapter 19.

Historical note: Rutherford achieved the first artificial nuclear reaction in 1919, observing protons emitted when alpha particles struck nitrogen gas. He correctly identified the reaction as ${}^{14}\text{N}(\alpha, p){}^{17}\text{O}$ — the transmutation of nitrogen into oxygen. This was the realization of the alchemist's dream, albeit on a single-atom scale and in the wrong direction (nitrogen to oxygen, not lead to gold). Cockcroft and Walton achieved the first accelerator-induced reaction in 1932: ${}^{7}\text{Li}(p, \alpha){}^{4}\text{He}$, splitting lithium with protons accelerated to 700 keV — a landmark that earned them the 1951 Nobel Prize.

17.2 Conservation Laws in Nuclear Reactions

Every nuclear reaction must conserve the following quantities:

17.2.1 Energy and Linear Momentum

Total relativistic energy and three-momentum are conserved:

$$E_a + E_b = E_c + E_d, \qquad \mathbf{p}_a + \mathbf{p}_b = \mathbf{p}_c + \mathbf{p}_d$$

In the laboratory frame where the target is at rest ($\mathbf{p}_a = 0$), the projectile carries all the initial momentum.

17.2.2 Electric Charge

$$Z_a + Z_b = Z_c + Z_d$$

Charge conservation is exact and absolute.

17.2.3 Baryon Number

$$A_a + A_b = A_c + A_d$$

Baryon number (mass number $A$) is conserved in all nuclear reactions. (Baryon number violation has never been observed experimentally, though it is predicted at extremely high energies in some grand unified theories.)

17.2.4 Lepton Number

In reactions involving beta decay or neutrinos, lepton number $L$ is conserved. For purely hadronic reactions (no leptons in or out), this is automatically satisfied.

17.2.5 Angular Momentum and Parity

Total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$ (orbital + intrinsic spin) is conserved. Parity is conserved in strong and electromagnetic interactions but violated in weak interactions (beta decay).

Example: Is the reaction ${}^{16}\text{O} + {}^{16}\text{O} \to {}^{31}\text{S} + p$ allowed by conservation laws?

• Charge: $8 + 8 = 16 + 1$ $\checkmark$
• Baryon number: $16 + 16 = 31 + 1$ $\checkmark$

The reaction is allowed. Whether it actually occurs at a given energy depends on the dynamics — the Q-value and the cross section.

17.2.6 Isospin

The strong nuclear force is approximately charge-independent — it does not distinguish between protons and neutrons. This symmetry is formalized through isospin $T$ (introduced in Chapter 2). In reactions mediated purely by the strong force, the total isospin and its third component $T_3$ are conserved. For example, the reaction ${}^{14}\text{C}(p, n){}^{14}\text{N}^*$ can only populate states of ${}^{14}\text{N}$ that satisfy the isospin selection rule.

Isospin conservation is approximate (broken by the Coulomb interaction), but it provides powerful selection rules for nuclear reactions. A classic example: the reaction $d + d \to {}^{4}\text{He} + \pi^0$ is isospin-forbidden because the deuteron has $T = 0$ and ${}^{4}\text{He}$ has $T = 0$, so the initial state has $T = 0$ and the final state would need the pion to carry $T = 1$ — violating isospin conservation.

17.2.7 Summary Table of Conservation Laws

The conservation laws serve as a first filter: if a proposed reaction violates any of the exact conservation laws (energy, momentum, charge, baryon number), it is absolutely forbidden. If it violates only approximate conservation laws (isospin), it may proceed but with a reduced cross section.

17.3 The Q-Value

17.3.1 Definition

The Q-value of a reaction is the difference between the total rest mass energy of the initial and final states:

$$Q = (M_a + M_b - M_c - M_d)\,c^2$$

where $M_i$ are the nuclear (or atomic) rest masses. Equivalently, using the kinetic energies $T_i$:

$$Q = T_c + T_d - T_a - T_b$$

In the laboratory frame with the target at rest ($T_a = 0$):

$$Q = T_c + T_d - T_b$$

The Q-value represents the net kinetic energy released (or absorbed) in the reaction.

17.3.2 Exothermic vs. Endothermic Reactions

• Exothermic ($Q > 0$): The products have less rest mass than the reactants. Kinetic energy is released. The reaction can proceed at any projectile energy (though the cross section may be negligibly small at very low energies due to the Coulomb barrier).

• Endothermic ($Q < 0$): The products have more rest mass than the reactants. Kinetic energy is absorbed. The reaction has a threshold energy below which it cannot occur.

17.3.3 Calculating Q-Values from Atomic Masses

In practice, Q-values are calculated from atomic masses $M_{\text{atom}}$ (which include electron masses) rather than nuclear masses. The electron masses cancel for reactions that conserve charge, because the same number of electrons appears on both sides. Using the Atomic Mass Evaluation (AME2020) values:

Example 1: ${}^{12}\text{C}(d, p){}^{13}\text{C}$

$$Q = \bigl[M({}^{12}\text{C}) + M(d) - M({}^{1}\text{H}) - M({}^{13}\text{C})\bigr]\,c^2$$

Using AME2020 atomic mass excesses $\Delta = (M - A\,u)\,c^2$ in MeV: - $\Delta({}^{12}\text{C}) = 0.000\,\text{MeV}$ (definition of the atomic mass unit) - $\Delta(d) = 13.136\,\text{MeV}$ - $\Delta({}^{1}\text{H}) = 7.289\,\text{MeV}$ - $\Delta({}^{13}\text{C}) = 3.125\,\text{MeV}$

$$Q = \Delta({}^{12}\text{C}) + \Delta(d) - \Delta({}^{1}\text{H}) - \Delta({}^{13}\text{C}) = 0.000 + 13.136 - 7.289 - 3.125 = +2.722\,\text{MeV}$$

This reaction is exothermic: the products are lighter than the reactants, and 2.722 MeV of kinetic energy is released.

Example 2: ${}^{12}\text{C}(p, n){}^{12}\text{N}$

$$Q = \Delta({}^{12}\text{C}) + \Delta({}^{1}\text{H}) - \Delta(n) - \Delta({}^{12}\text{N})$$

$$= 0.000 + 7.289 - 8.071 - 17.338 = -18.120\,\text{MeV}$$

This reaction is strongly endothermic: a proton must bring at least enough kinetic energy to create the mass difference before ${}^{12}\text{N}$ can be produced.

Connection to binding energy: The Q-value can also be written as $Q = B_c + B_d - B_a - B_b$, where $B$ is the total binding energy. A reaction that produces more tightly bound products (higher $B$) is exothermic.

17.3.4 Q-Values and Separation Energies

The Q-value of a capture reaction is directly related to the separation energy of the captured particle. For neutron capture $a(n, \gamma)b$, the Q-value equals the neutron separation energy of the product:

$$Q(n, \gamma) = S_n(b) = [M({}^{A-1}X) + M_n - M({}^{A}X)]\,c^2$$

Since most nuclei have neutron separation energies of 5--10 MeV (comparable to the binding energy per nucleon), most $(n, \gamma)$ reactions are exothermic with $Q \approx 6$--$8\,\text{MeV}$.

Similarly, for $(d, p)$ stripping reactions:

$$Q(d, p) = S_n(\text{product}) - B_d$$

where $B_d = 2.224\,\text{MeV}$ is the deuteron binding energy. Since $S_n \sim 6$--$8\,\text{MeV} > B_d$, most $(d, p)$ reactions are exothermic. This is why $(d, p)$ reactions are such versatile spectroscopic tools — they work at convenient beam energies with positive Q-values for almost all targets.

Example 3: ${}^{7}\text{Li}(p, \alpha){}^{4}\text{He}$ (Cockcroft-Walton reaction)

$$Q = \Delta({}^{7}\text{Li}) + \Delta({}^{1}\text{H}) - 2\Delta({}^{4}\text{He}) = 14.908 + 7.289 - 2(2.425) = +17.347\,\text{MeV}$$

This very large positive Q-value is why the two alpha particles from this reaction were easily detected in the original 1932 Cockcroft-Walton experiment — each alpha received about 8.7 MeV of kinetic energy, far more than the 700 keV input. The experiment demonstrated that $E = mc^2$ was quantitatively correct: the mass deficit of 0.0186 u corresponded to the observed kinetic energy release of 17.3 MeV.

17.3.5 The Q-Value Equation

For a two-body reaction $a(b,c)d$ in the laboratory frame (target $a$ at rest), conservation of energy and momentum gives the Q-value equation relating $Q$ to the measured quantities $T_b$ (beam energy), $T_c$ (ejectile energy), and $\theta_{\text{lab}}$ (ejectile angle):

$$Q = T_c\left(1 + \frac{M_c}{M_d}\right) - T_b\left(1 - \frac{M_b}{M_d}\right) - \frac{2}{M_d}\sqrt{M_b M_c T_b T_c}\,\cos\theta_{\text{lab}}$$

This is a key experimental equation: by measuring $T_c$ and $\theta_{\text{lab}}$ for known $T_b$ and masses, one determines $Q$ and hence identifies the reaction.

Experimental technique: Q-value spectroscopy. In practice, experimentalists measure the ejectile energy $T_c$ at a fixed angle $\theta_{\text{lab}}$ for a known beam energy $T_b$. Each discrete state of the residual nucleus $d$ corresponds to a different Q-value (the ground-state Q-value minus the excitation energy $E^*$), and hence a different ejectile energy. The energy spectrum of the ejectiles therefore displays a series of peaks, each corresponding to a populated state in $d$. This is the basis of nuclear spectroscopy via transfer reactions — a technique explored in detail in Chapter 19.

Worked numerical example: the Q-value equation in action. In a ${}^{12}\text{C}(d, p){}^{13}\text{C}$ experiment at $T_d = 15\,\text{MeV}$, a proton is detected at $\theta_{\text{lab}} = 30°$ with kinetic energy $T_p = 16.8\,\text{MeV}$.

$$Q = 16.8\left(1 + \frac{1}{13}\right) - 15\left(1 - \frac{2}{13}\right) - \frac{2}{13}\sqrt{2 \times 1 \times 15 \times 16.8}\cos 30°$$

$$= 16.8 \times 1.077 - 15 \times 0.846 - 0.154 \times 22.45 \times 0.866$$

$$= 18.09 - 12.69 - 2.99 = +2.41\,\text{MeV}$$

This is close to the known ground-state Q-value of $+2.722\,\text{MeV}$; the small discrepancy (in this simplified calculation) reflects the fact that the proton energy places it near an excited state of ${}^{13}\text{C}$.

17.4 Threshold Energy

17.4.1 The Threshold Condition

For an endothermic reaction ($Q < 0$), the projectile must supply enough kinetic energy not only to create the mass difference but also to conserve momentum. The threshold energy $E_{\text{th}}$ is the minimum laboratory kinetic energy of the projectile for which the reaction is kinematically allowed.

At threshold, all products are created at rest in the center-of-mass (CM) frame — they move together as a single unit in the laboratory frame. Applying conservation of energy and momentum:

$$E_{\text{th}} = -Q \cdot \frac{M_a + M_b + M_c + M_d}{2M_a}$$

For the common case where the product masses are close to the reactant masses ($M_c + M_d \approx M_a + M_b$), this simplifies to:

$$E_{\text{th}} \approx -Q\left(1 + \frac{M_b}{M_a}\right)$$

The threshold always exceeds $|Q|$ because some kinetic energy must go into center-of-mass motion to conserve momentum.

17.4.2 Derivation

In the laboratory frame (target at rest), the total four-momentum is:

$$p^{\mu}_{\text{total}} = \bigl((M_a + T_b/c^2)c, \, \mathbf{p}_b\bigr) \quad \text{(using } M_a c^2 + T_b \text{ for the total energy)}$$

At threshold, all products are at rest in the CM frame, so the invariant mass of the final state equals $M_c + M_d$ (the minimum possible). The invariant mass squared is a Lorentz scalar:

$$s = \left(\sum E_i\right)^2 - \left(\sum \mathbf{p}_i c\right)^2 c^{-2}$$

In the lab frame:

$$s\,c^4 = (M_a c^2 + M_b c^2 + T_b)^2 - 2M_b c^2 T_b - T_b^2$$

Using $|\mathbf{p}_b|^2 c^2 = T_b^2 + 2M_b c^2 T_b$ and expanding:

$$s\,c^4 = M_a^2 c^4 + M_b^2 c^4 + 2M_a c^2(M_b c^2 + T_b)$$

At threshold, $\sqrt{s}\,c^2 = (M_c + M_d)c^2$, so:

$$(M_c + M_d)^2 c^4 = M_a^2 c^4 + M_b^2 c^4 + 2M_a c^2(M_b c^2 + T_{\text{th}})$$

Solving for $T_{\text{th}}$:

$$\boxed{T_{\text{th}} = \frac{(M_c + M_d)^2 - (M_a + M_b)^2}{2M_a}\,c^2 = -Q\,\frac{(M_c + M_d)^2 - (M_a + M_b)^2}{2M_a\,Q}\,c^2}$$

Since $Q = (M_a + M_b - M_c - M_d)c^2$, we can factor:

$$(M_c + M_d)^2 - (M_a + M_b)^2 = -\frac{Q}{c^2}\bigl[(M_a + M_b) + (M_c + M_d)\bigr]$$

Therefore:

$$\boxed{T_{\text{th}} = -Q\cdot\frac{M_a + M_b + M_c + M_d}{2M_a}}$$

Example: For ${}^{12}\text{C}(p, n){}^{12}\text{N}$ with $Q = -18.120\,\text{MeV}$:

$$T_{\text{th}} = 18.120 \times \frac{12 + 1 + 1 + 12}{2 \times 12} = 18.120 \times \frac{26}{24} = 19.63\,\text{MeV}$$

The proton must have at least 19.63 MeV kinetic energy — about 1.5 MeV more than $|Q|$ — to produce ${}^{12}\text{N}$.

17.5 Center-of-Mass and Laboratory Frames

17.5.1 Why Two Frames?

Nuclear reactions are performed in the laboratory frame (target at rest, projectile moving), but the physics is simplest in the center-of-mass (CM) frame (total momentum zero). Experimentalists measure in the lab; theorists calculate in the CM. Converting between the two is essential.

17.5.2 CM Frame Definitions

The center-of-mass velocity is:

$$\mathbf{v}_{\text{CM}} = \frac{M_b \mathbf{v}_b}{M_a + M_b}$$

where $\mathbf{v}_b$ is the projectile velocity in the lab. The reduced mass is:

$$\mu = \frac{M_a M_b}{M_a + M_b}$$

The kinetic energy available for the reaction in the CM frame is:

$$T_{\text{CM}} = \frac{M_a}{M_a + M_b}\,T_{\text{lab}} = \frac{1}{2}\mu v_{\text{rel}}^2$$

where $v_{\text{rel}} = |\mathbf{v}_b - \mathbf{v}_a| = v_b$ (since the target is at rest). The fraction of laboratory kinetic energy available in the CM frame decreases as the projectile-to-target mass ratio increases — a crucial consideration in accelerator design.

Numerical insight: For a proton ($M_b = 1\,u$) hitting a lead target ($M_a = 208\,u$), $T_{\text{CM}}/T_{\text{lab}} = 208/209 = 0.995$ — almost all the lab energy is available. For symmetric collisions ($M_a = M_b$), only half the lab energy is available in the CM. This is why collider experiments are far more energy-efficient than fixed-target experiments for creating massive particles.

17.5.3 Invariant Mass and the Mandelstam Variable $s$

The invariant mass $\sqrt{s}$ of the system is a Lorentz scalar that takes the same value in all frames:

$$s = (E_a + E_b)^2/c^4 - |\mathbf{p}_a + \mathbf{p}_b|^2/c^2$$

In the lab frame (non-relativistic):

$$s\,c^4 \approx (M_a + M_b)^2c^4 + 2M_a c^2 T_{\text{lab}}$$

In the CM frame:

$$\sqrt{s}\,c^2 = (M_a + M_b)c^2 + T_{\text{CM}}$$

The invariant mass determines which exit channels are energetically accessible: a channel with product masses $M_c + M_d$ is open only if $\sqrt{s}\,c^2 \geq (M_c + M_d)c^2$.

17.5.4 Angle Transformation: Lab to CM

The relationship between the CM scattering angle $\theta_{\text{CM}}$ and the lab scattering angle $\theta_{\text{lab}}$ for the ejectile $c$ is:

$$\tan\theta_{\text{lab}} = \frac{\sin\theta_{\text{CM}}}{\cos\theta_{\text{CM}} + \eta}$$

where the parameter $\eta$ is the ratio of the CM velocity to the velocity of the ejectile in the CM frame:

$$\eta = \frac{v_{\text{CM}}}{v_c^{\text{CM}}} = \frac{M_b}{M_d}\sqrt{\frac{M_c T_b}{M_a(Q + T_{\text{CM}})}} \cdot \frac{1}{\sqrt{1 + Q/T_{\text{CM}}}}$$

For elastic scattering ($Q = 0$, $c = b$, $d = a$), this simplifies to:

$$\eta = \frac{M_b}{M_a}$$

Three important cases: 1. $\eta < 1$ (light projectile, heavy target): $\theta_{\text{lab}}$ ranges from $0$ to $\pi$. Each lab angle corresponds to a single CM angle. This is the common situation in nuclear physics experiments.

2. $\eta = 1$ (equal masses): $\theta_{\text{lab}}^{\max} = \pi/2$. The two particles always emerge at right angles in the lab.

3. $\eta > 1$ (heavy projectile, light target): $\theta_{\text{lab}}$ has a maximum value $< \pi/2$, and two CM angles correspond to each lab angle (forward and backward CM scattering both produce the same lab angle). This is called kinematic focusing.

17.5.5 Jacobian for Cross Section Transformation

The differential cross section transforms between frames according to:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{lab}} = \left(\frac{d\sigma}{d\Omega}\right)_{\text{CM}} \cdot \frac{(1 + 2\eta\cos\theta_{\text{CM}} + \eta^2)^{3/2}}{|1 + \eta\cos\theta_{\text{CM}}|}$$

This Jacobian factor accounts for the change in solid angle element $d\Omega$ between the two frames. It must be applied whenever comparing theoretical (CM frame) predictions to experimental (lab frame) data.

17.5.6 Relativistic Kinematics

For beam energies above about 100 MeV per nucleon, relativistic effects become important. The fully relativistic kinematics uses four-vectors. The total four-momentum in the lab frame is:

$$p^{\mu}_{\text{tot}} = (E_a/c + E_b/c,\, \mathbf{p}_b)$$

where $E_a = M_a c^2$ (target at rest) and $E_b = \sqrt{(p_b c)^2 + (M_b c^2)^2}$. The invariant mass squared is:

$$s = \frac{1}{c^4}(E_a + E_b)^2 - \frac{|\mathbf{p}_b|^2}{c^2} = M_a^2 + M_b^2 + \frac{2M_a E_b}{c^2}$$

For a fixed-target experiment, $\sqrt{s}$ grows only as $\sqrt{E_b}$ at high energies (most of the lab energy goes into CM motion). For a collider ($\mathbf{p}_a = -\mathbf{p}_b$ in the lab), $\sqrt{s} \propto E_b$ — the full energy is available. This is why colliders are essential for high-energy particle physics, but fixed-target geometry dominates nuclear physics, where the beam energies are lower and thick targets provide high luminosity.

Worked example: $\sqrt{s}$ for a FRIB experiment. At the Facility for Rare Isotope Beams (FRIB), a ${}^{132}\text{Sn}$ beam at 200 MeV/nucleon ($T_{\text{lab}} = 200 \times 132 = 26{,}400\,\text{MeV}$) strikes a liquid hydrogen target. The invariant mass:

$$\sqrt{s}\,c^2 \approx \sqrt{(M_a + M_b)^2c^4 + 2M_a c^2 \cdot T_{\text{lab}}}$$

With $M_a c^2 \approx 938\,\text{MeV}$ (proton) and $M_b c^2 \approx 132 \times 931 \approx 122{,}900\,\text{MeV}$:

$$\sqrt{s}\,c^2 \approx \sqrt{(938 + 122900)^2 + 2 \times 938 \times 26400} \approx 124{,}040\,\text{MeV}$$

The CM kinetic energy is $\sqrt{s}\,c^2 - (M_a + M_b)c^2 \approx 202\,\text{MeV}$ — most of the 26.4 GeV of lab kinetic energy goes into CM motion, with only 202 MeV available for the reaction. This illustrates why even "intermediate energy" beams at rare-isotope facilities operate in a regime where careful relativistic kinematics is necessary.

17.6 The Cross Section

17.6.1 The Concept

Imagine firing a beam of particles at a thin target foil. Some beam particles pass through unscattered; some scatter into various angles; some initiate reactions that produce new particles. The cross section $\sigma$ quantifies the probability of a particular process occurring.

Consider a beam of intensity $I$ (particles per second per unit area) incident on a thin target containing $n$ atoms per unit area (number density $\rho N_A / A$ times thickness $t$). The reaction rate $R$ (events per second) is:

$$R = I \cdot \sigma \cdot n$$

Solving for $\sigma$:

$$\sigma = \frac{R}{I \cdot n}$$

The cross section has dimensions of area and is measured in barns:

$$1\,\text{barn} = 10^{-24}\,\text{cm}^2 = 10^{-28}\,\text{m}^2 = 100\,\text{fm}^2$$

The name "barn" originated during the Manhattan Project: compared to the tiny nuclei they were trying to hit, the uranium fission cross section (of order 1 barn) was "as big as a barn." The geometric cross-sectional area of a nucleus with radius $R = r_0 A^{1/3}$ ($r_0 \approx 1.2\,\text{fm}$) is $\pi R^2 \approx 45 A^{2/3}\,\text{mb}$, giving roughly 1 barn for $A \sim 100$. But actual cross sections can be enormously different from this geometric estimate:

The cross section is not a physical area of the target — it is an effective area that encodes the full quantum-mechanical interaction dynamics.

17.6.2 Differential Cross Section

The differential cross section $d\sigma/d\Omega$ gives the probability of scattering into a particular solid angle element $d\Omega = \sin\theta\,d\theta\,d\phi$:

$$\frac{dR}{d\Omega} = I \cdot n \cdot \frac{d\sigma}{d\Omega}$$

The total cross section is obtained by integrating over all solid angles:

$$\sigma = \int \frac{d\sigma}{d\Omega}\,d\Omega = \int_0^{2\pi}d\phi \int_0^{\pi}\frac{d\sigma}{d\Omega}\sin\theta\,d\theta$$

For azimuthally symmetric scattering (unpolarized beam, unpolarized target):

$$\sigma = 2\pi\int_0^{\pi}\frac{d\sigma}{d\Omega}\sin\theta\,d\theta$$

17.6.3 Measuring Cross Sections

A typical scattering experiment measures the number of particles $\Delta N$ detected in a detector of solid angle $\Delta\Omega$ at angle $\theta$:

$$\Delta N = N_{\text{beam}} \cdot n_{\text{target}} \cdot \frac{d\sigma}{d\Omega}\bigg|_{\theta} \cdot \Delta\Omega$$

where $N_{\text{beam}}$ is the total number of beam particles and $n_{\text{target}}$ is the number of target atoms per unit area. The detector solid angle is $\Delta\Omega = A_{\text{det}} / r^2$, where $A_{\text{det}}$ is the detector area and $r$ is the distance from the target.

Absolute cross section measurements require knowing $N_{\text{beam}}$, $n_{\text{target}}$, $\Delta\Omega$, and the detector efficiency — each with its own systematic uncertainties. Typical absolute uncertainties are 5--10%. Relative angular distributions (ratios of cross sections at different angles) can be measured to 1--2%.

Practical note: transmission measurements. An alternative to angular-distribution measurements is the transmission experiment: measure the attenuation of a beam passing through a thick target. If the incident intensity is $I_0$ and the transmitted intensity is $I$, then $I = I_0\,e^{-n\sigma_{\text{tot}} x}$, where $n$ is the number density and $x$ is the target thickness. This gives the total cross section (elastic + reaction) directly. Total cross section measurements by transmission are the most accurate absolute cross section measurements, with uncertainties as low as 0.1% for neutrons.

17.6.4 Energy Dependence of Cross Sections

Cross sections vary enormously with beam energy. Several general features:

1. $1/v$ law at low energies: For exothermic neutron-induced reactions, $\sigma \propto 1/v \propto 1/\sqrt{E}$ at low energies. This is because the neutron spends more time near the target at lower velocities. More precisely, the reaction probability per encounter is roughly constant, and the time the neutron spends near the nucleus scales as $1/v$. The $1/v$ law is observed for ${}^{10}\text{B}(n, \alpha)$, ${}^{6}\text{Li}(n, t)$, and most $(n, \gamma)$ reactions far from resonances.

2. Resonances: Sharp peaks in $\sigma(E)$ occur when the compound nucleus (Chapter 18) is excited to a quasi-bound state. The ${}^{238}\text{U}(n, \gamma)$ cross section, for example, shows hundreds of narrow resonances between 1 eV and 10 keV. The widths of these resonances ($\Gamma \sim 0.01$--$1\,\text{eV}$ for neutron-induced reactions on heavy nuclei) reflect the lifetime of the compound nucleus state: $\tau = \hbar/\Gamma$.

3. Coulomb barrier suppression: For charged-particle reactions, the cross section is exponentially suppressed at energies below the Coulomb barrier: $\sigma \propto \exp(-2\pi\eta)$, where $\eta = Z_a Z_b e^2/(4\pi\epsilon_0 \hbar v)$ is the Sommerfeld parameter. The Coulomb barrier height is approximately $V_C = k Z_a Z_b e^2 / (r_0(A_a^{1/3} + A_b^{1/3})) \approx 1.44 Z_a Z_b / (1.2(A_a^{1/3} + A_b^{1/3}))\,\text{MeV}$, which ranges from about 1 MeV for $p + {}^{6}\text{Li}$ to about 200 MeV for ${}^{92}\text{Mo} + {}^{92}\text{Mo}$. Below the barrier, cross sections fall by orders of magnitude per MeV of energy decrease.

4. High-energy limit: At very high energies, the total reaction cross section approaches the geometric cross section $\sigma_R \approx \pi R^2 \approx \pi r_0^2(A_a^{1/3} + A_b^{1/3})^2$. For nucleon-nucleus scattering at $E > 100\,\text{MeV}$, the total cross section is approximately $\sigma_{\text{tot}} \approx 2\pi R^2$ (the factor of 2 from shadow scattering, discussed in Section 17.8).

5. The astrophysical S-factor: For charged-particle reactions at astrophysical energies (far below the Coulomb barrier), the steep energy dependence of the Coulomb penetration makes $\sigma(E)$ vary by many orders of magnitude. It is conventional to factor out the penetration probability and define the astrophysical S-factor:

$$S(E) = \sigma(E) \cdot E \cdot \exp(2\pi\eta)$$

The S-factor is a slowly varying function of energy, making it possible to extrapolate measured cross sections to the very low energies relevant in stellar interiors ($E \sim 1$--$100\,\text{keV}$). This is essential for nuclear astrophysics (Chapter 22).

A sense of scale. The dynamic range of nuclear cross sections is staggering. At one extreme, the ${}^{157}\text{Gd}(n, \gamma)$ cross section at thermal energies is $254{,}000\,\text{b}$ — about $6 \times 10^6$ times the geometric cross section of the gadolinium nucleus. At the other extreme, the solar $pp$ reaction $p + p \to d + e^+ + \nu_e$ has a cross section of approximately $10^{-47}\,\text{cm}^2 = 10^{-23}\,\text{b}$ — so small that a proton at the center of the Sun will wait, on average, about 10 billion years before fusing. Yet this tiny cross section powers the Sun.

17.7 Rutherford Scattering: Full Derivation

The Rutherford scattering cross section — the differential cross section for elastic Coulomb scattering of a point charge off a point charge — is one of the most important results in nuclear and atomic physics. We derive it here in full, first classically, then confirm the quantum-mechanical result.

17.7.1 Classical Coulomb Scattering

Consider a projectile of charge $z_1 e$ and mass $m_1$ scattering off a target nucleus of charge $z_2 e$ and mass $m_2 \gg m_1$ (we will correct for finite target mass afterward). The Coulomb potential is:

$$V(r) = \frac{k z_1 z_2 e^2}{r}, \qquad k = \frac{1}{4\pi\epsilon_0}$$

We define the Coulomb parameter:

$$a = \frac{k z_1 z_2 e^2}{2E_{\text{CM}}} = \frac{z_1 z_2 e^2}{4\pi\epsilon_0 \cdot 2E_{\text{CM}}}$$

where $E_{\text{CM}} = \frac{1}{2}\mu v^2$ is the kinetic energy in the CM frame and $\mu$ is the reduced mass. The parameter $a$ is half the distance of closest approach for a head-on collision ($b = 0$): the projectile reaches a minimum distance $d_0 = 2a$ before the Coulomb repulsion turns it around.

17.7.2 Orbit Equation

The projectile moves in a plane (by conservation of angular momentum $L = \mu v b$, where $b$ is the impact parameter). Using the orbit equation for a $1/r$ potential (the same Kepler problem as planetary orbits, but with a repulsive force), the trajectory is a hyperbola:

$$\frac{1}{r} = \frac{1}{p}\bigl(-1 + \varepsilon\cos\phi\bigr)$$

where $p = L^2/(\mu k z_1 z_2 e^2) = b^2/a$ is the semi-latus rectum and $\varepsilon = \sqrt{1 + (b/a)^2}$ is the eccentricity (always $> 1$ for a repulsive potential — a hyperbolic orbit).

The asymptotic deflection angle $\Theta$ (the total change in direction of the projectile, measured in the CM frame) is related to the eccentricity by:

$$\cot\frac{\Theta}{2} = \varepsilon \sin\alpha = \frac{b}{a}\sqrt{1 + (b/a)^2}\cdot\frac{1}{\sqrt{1+(b/a)^2}} = \frac{b}{a}$$

Wait — let us do this carefully. The asymptotes of the hyperbola make an angle $\alpha$ with the axis of symmetry, where $\cos\alpha = 1/\varepsilon$. The deflection angle is $\Theta = \pi - 2\alpha$, so:

$$\alpha = \frac{\pi - \Theta}{2}$$

and thus:

$$\cos\alpha = \cos\left(\frac{\pi - \Theta}{2}\right) = \sin\frac{\Theta}{2} = \frac{1}{\varepsilon}$$

Since $\varepsilon = \sqrt{1 + b^2/a^2}$, we get:

$$\sin^2\frac{\Theta}{2} = \frac{1}{1 + b^2/a^2}$$

Solving for $b$:

$$\boxed{b = a\cot\frac{\Theta}{2}}$$

This is the fundamental relationship between the impact parameter $b$ and the scattering angle $\Theta$ for Coulomb scattering. Large impact parameters ($b \gg a$) give small deflections ($\Theta \to 0$); head-on collisions ($b = 0$) give $\Theta = \pi$ (back-scattering).

17.7.3 From Impact Parameter to Cross Section

The key insight connecting the orbit to the cross section is purely geometric. Particles incident with impact parameters between $b$ and $b + db$ scatter into angles between $\Theta$ and $\Theta + d\Theta$. By axial symmetry, the annular ring of area $d\sigma = 2\pi b\,|db|$ maps to the solid angle element $d\Omega = 2\pi\sin\Theta\,|d\Theta|$. The differential cross section is:

$$\frac{d\sigma}{d\Omega} = \frac{b}{\sin\Theta}\left|\frac{db}{d\Theta}\right|$$

From $b = a\cot(\Theta/2)$:

$$\frac{db}{d\Theta} = -\frac{a}{2\sin^2(\Theta/2)}$$

Therefore:

$$\frac{d\sigma}{d\Omega} = \frac{a\cot(\Theta/2)}{\sin\Theta} \cdot \frac{a}{2\sin^2(\Theta/2)}$$

Using $\sin\Theta = 2\sin(\Theta/2)\cos(\Theta/2)$ and $\cot(\Theta/2) = \cos(\Theta/2)/\sin(\Theta/2)$:

$$\frac{d\sigma}{d\Omega} = \frac{a^2\cos(\Theta/2)/\sin(\Theta/2)}{2\sin(\Theta/2)\cos(\Theta/2)} \cdot \frac{1}{2\sin^2(\Theta/2)} = \frac{a^2}{4\sin^4(\Theta/2)}$$

Substituting $a = k z_1 z_2 e^2 / (2E_{\text{CM}})$:

$$\boxed{\left(\frac{d\sigma}{d\Omega}\right)_{\text{Ruth}} = \left(\frac{z_1 z_2 e^2}{4 \cdot 4\pi\epsilon_0 E_{\text{CM}}}\right)^2 \frac{1}{\sin^4(\Theta/2)} = \left(\frac{a}{2}\right)^2 \frac{1}{\sin^4(\Theta/2)}}$$

This is the Rutherford scattering formula. Writing it with the more compact notation $a_0 = k z_1 z_2 e^2 / (4 E_{\text{CM}})$:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{Ruth}} = a_0^2 \cdot \frac{1}{\sin^4(\Theta/2)}$$

17.7.4 Key Features of the Rutherford Cross Section

1. Strong forward peaking: The $1/\sin^4(\Theta/2)$ factor diverges as $\Theta \to 0$, meaning the cross section is enormously enhanced at small angles. This reflects the long range of the Coulomb force — even distant encounters (large $b$) cause small deflections.

2. Energy dependence: $d\sigma/d\Omega \propto 1/E_{\text{CM}}^2$. Higher-energy projectiles are harder to deflect, reducing the cross section at all angles.

3. Charge dependence: $d\sigma/d\Omega \propto (z_1 z_2)^2$. Heavy nuclei scatter more strongly. This is why Rutherford used gold ($Z = 79$) — the large $Z$ maximized the scattering signal.

4. Infinite total cross section: Integrating over all solid angles gives $\sigma_{\text{total}} = \infty$, because the $1/\sin^4(\Theta/2)$ singularity at $\theta = 0$ is not integrable. Physically, this reflects the infinite range of the Coulomb force — no matter how large $b$ is, there is always some (tiny) deflection.

5. No dependence on nuclear structure: The Rutherford formula treats both projectile and target as point charges. Deviations from Rutherford scattering at large angles (where the projectile approaches closely enough to "feel" the nuclear interior) are precisely how nuclear sizes and shapes are measured.

17.7.5 Quantum-Mechanical Confirmation

Remarkably, the exact quantum-mechanical calculation of Coulomb scattering (first performed by Gordon in 1928 and Mott in 1928) gives the same result as the classical derivation for distinguishable, spinless particles. This coincidence occurs because the Coulomb potential is a $1/r$ potential, for which the classical and quantum cross sections happen to agree — a special property not shared by other potential forms.

The quantum scattering amplitude for Coulomb scattering is:

$$f_C(\Theta) = -\frac{a}{2\sin^2(\Theta/2)}\,\exp\!\bigl[-i\,\eta\ln\sin^2(\Theta/2) + 2i\sigma_0\bigr]$$

where $\eta = a/\lambdabar$ is the Sommerfeld parameter ($\lambdabar = \hbar/\mu v$) and $\sigma_0 = \arg\Gamma(1 + i\eta)$ is the Coulomb phase shift. The squared modulus $|f_C|^2 = a^2/(4\sin^4(\Theta/2))$ recovers the Rutherford formula exactly.

For identical particles (e.g., $\alpha$--$\alpha$ scattering), the quantum result differs from the classical one because the scattering amplitude must be symmetrized:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{Mott}} = |f_C(\Theta) + f_C(\pi - \Theta)|^2$$

This produces interference oscillations superimposed on the Rutherford envelope — the Mott scattering pattern, which was historically important as evidence for the quantum nature of nuclear scattering.

17.7.6 Finite Target Mass Correction

For a target of finite mass $M_a$, the scattering angle $\theta_{\text{lab}}$ in the laboratory differs from the CM angle $\Theta$. The Rutherford cross section in the lab frame becomes:

$$\left(\frac{d\sigma}{d\Omega}\right)_{\text{lab}} = \left(\frac{a}{2}\right)^2 \frac{4}{\sin^4\theta_{\text{lab}}}\cdot\frac{\left[\cos\theta_{\text{lab}} + \sqrt{1 - (M_b/M_a)^2\sin^2\theta_{\text{lab}}}\right]^2}{\sqrt{1 - (M_b/M_a)^2\sin^2\theta_{\text{lab}}}}$$

For $M_a \gg M_b$, this reduces to the standard formula with $\theta_{\text{lab}} \approx \Theta$. For protons scattering off light nuclei, the correction is significant and must be included.

17.7.7 Numerical Example: When Does Rutherford Break Down?

For alpha particles scattering off ${}^{40}\text{Ca}$ ($Z = 20$, $R \approx 1.25 \times 40^{1/3} \approx 4.3\,\text{fm}$), the Rutherford formula breaks down when the distance of closest approach $d_0$ equals the nuclear radius. For back-scattering ($\theta = 180°$):

$$d_0 = \frac{2 \times 20 \times 1.440}{E_{\text{CM}}} = \frac{57.6}{E_{\text{CM}}}\,\text{fm}$$

Setting $d_0 = R$: $E_{\text{CM}} \approx 57.6/4.3 \approx 13.4\,\text{MeV}$, corresponding to $E_{\text{lab}} \approx 13.4 \times (40 + 4)/40 \approx 14.7\,\text{MeV}$.

Experimentally, deviations from the Rutherford formula at $\theta = 180°$ begin at $E_{\text{lab}} \approx 14\,\text{MeV}$ for this system, in excellent agreement with this estimate. Below this energy, Rutherford scattering provides an absolute cross section normalization — a technique routinely exploited in nuclear physics experiments. The angular distribution is measured relative to Rutherford scattering at forward angles (where the Rutherford formula is guaranteed to hold) and then normalized absolutely.

17.8 Partial Wave Analysis

17.8.1 From Plane Waves to Angular Momentum

In quantum scattering theory, the incident beam is represented as a plane wave $e^{ikz}$ (choosing the beam direction along $\hat{z}$). Far from the scattering center, the total wavefunction has the asymptotic form:

$$\psi(\mathbf{r}) \xrightarrow{r \to \infty} e^{ikz} + f(\theta)\frac{e^{ikr}}{r}$$

where $f(\theta)$ is the scattering amplitude and the differential cross section is $d\sigma/d\Omega = |f(\theta)|^2$.

The plane wave can be expanded in partial waves — components of definite orbital angular momentum $l$:

$$e^{ikz} = \sum_{l=0}^{\infty}(2l+1)\,i^l\,j_l(kr)\,P_l(\cos\theta)$$

where $j_l(kr)$ are spherical Bessel functions and $P_l(\cos\theta)$ are Legendre polynomials. Each term corresponds to a definite angular momentum $l\hbar$ about the scattering center. The key insight: for a central potential $V(r)$, angular momentum is conserved, so each partial wave scatters independently.

17.8.2 Phase Shifts

For each partial wave $l$, the effect of the potential is to introduce a phase shift $\delta_l$ in the asymptotic radial wavefunction. In the absence of the potential, the radial wavefunction for partial wave $l$ behaves asymptotically as:

$$u_l(r) \xrightarrow{r \to \infty} \sin\left(kr - \frac{l\pi}{2}\right)$$

With the potential:

$$u_l(r) \xrightarrow{r \to \infty} \sin\left(kr - \frac{l\pi}{2} + \delta_l\right)$$

The scattering amplitude is:

$$f(\theta) = \frac{1}{2ik}\sum_{l=0}^{\infty}(2l+1)\bigl(e^{2i\delta_l} - 1\bigr)P_l(\cos\theta) = \frac{1}{k}\sum_{l=0}^{\infty}(2l+1)\,e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta)$$

The total elastic cross section is:

$$\sigma_{\text{el}} = \frac{4\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2\delta_l$$

This is a fundamental result: each partial wave contributes at most $(2l+1) \cdot 4\pi/k^2$ to the cross section (the unitarity limit), achieved when $\delta_l = \pi/2$ (resonance condition).

17.8.3 Which Partial Waves Matter?

Classically, a partial wave $l$ corresponds to an impact parameter $b_l \approx (l + 1/2)/k$. Only partial waves with $b_l \lesssim R$ (the nuclear radius plus the range of the interaction) contribute significantly. This gives a maximum angular momentum:

$$l_{\max} \approx kR = \frac{pR}{\hbar} = \frac{\sqrt{2\mu E_{\text{CM}}}\cdot R}{\hbar}$$

At low energies ($kR \ll 1$), only $l = 0$ (s-wave) scattering is important, and the cross section is isotropic:

$$\sigma \approx \frac{4\pi}{k^2}\sin^2\delta_0$$

At high energies ($kR \gg 1$), many partial waves contribute, and the angular distribution becomes complex. For a 10 MeV neutron scattering off ${}^{208}\text{Pb}$:

$$l_{\max} \approx kR \approx \frac{\sqrt{2 \times 939 \times 10}\,\text{MeV}/c \times 7.1\,\text{fm}}{\hbar c} \approx \frac{137\,\text{MeV}/c \times 7.1\,\text{fm}}{197\,\text{MeV}\cdot\text{fm}} \approx 5$$

So about 6 partial waves ($l = 0$ through $l = 5$) contribute.

17.8.4 The Centrifugal Barrier

The reason higher partial waves are suppressed at low energies is the centrifugal barrier. The effective potential for partial wave $l$ is:

$$V_{\text{eff}}(r) = V(r) + \frac{\hbar^2 l(l+1)}{2\mu r^2}$$

The centrifugal term $\hbar^2 l(l+1)/(2\mu r^2)$ creates a barrier that the projectile must penetrate to reach the nuclear surface. The height of this barrier at the nuclear radius $R$ is:

$$V_{\text{cent}}(R) = \frac{\hbar^2 l(l+1)}{2\mu R^2}$$

For neutrons (no Coulomb barrier) scattering off ${}^{208}\text{Pb}$ ($R \approx 7.1\,\text{fm}$), the centrifugal barrier heights are:

At $E = 1\,\text{MeV}$, only $l = 0$ and $l = 1$ can classically overcome the centrifugal barrier. Higher partial waves can still contribute through quantum tunneling, but their contribution is exponentially suppressed.

For charged-particle reactions, the Coulomb barrier adds to the centrifugal barrier, further suppressing higher partial waves. The combined Coulomb-plus-centrifugal barrier determines the energy dependence of the reaction cross section — a critical input for nuclear astrophysics (Chapter 22).

17.8.5 Transmission Coefficients and Reaction Cross Sections

When absorption (nuclear reactions) is possible, the outgoing wave in each partial wave is reduced in amplitude. We define the S-matrix element $S_l = \eta_l e^{2i\delta_l}$, where $\eta_l$ ($0 \leq \eta_l \leq 1$) is the elasticity parameter:

• $\eta_l = 1$: pure elastic scattering (no absorption)
• $\eta_l = 0$: complete absorption (all flux in this partial wave is removed)

The transmission coefficient is $T_l = 1 - \eta_l^2$, representing the fraction of the incoming flux in partial wave $l$ that is absorbed (i.e., leads to reactions).

The reaction (absorption) cross section is:

$$\sigma_{\text{rxn}} = \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\,T_l = \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)(1 - |\eta_l|^2)$$

The elastic cross section becomes:

$$\sigma_{\text{el}} = \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)|1 - S_l|^2 = \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)|1 - \eta_l e^{2i\delta_l}|^2$$

The total cross section is:

$$\sigma_{\text{tot}} = \sigma_{\text{el}} + \sigma_{\text{rxn}} = \frac{2\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)(1 - \eta_l\cos 2\delta_l)$$

For a perfect absorber ($\eta_l = 0$ for $l \leq l_{\max}$, $\eta_l = 1$ for $l > l_{\max}$):

$$\sigma_{\text{rxn}} = \frac{\pi}{k^2}\sum_{l=0}^{l_{\max}}(2l+1) \approx \pi(l_{\max}/k)^2 \approx \pi R^2$$

But the elastic (shadow scattering) cross section is also $\approx \pi R^2$, giving a total cross section $\sigma_{\text{tot}} \approx 2\pi R^2$. The factor of 2 is the shadow scattering phenomenon: a perfectly absorbing sphere removes flux from the beam, and the resulting diffraction pattern has a forward peak that contributes an additional $\pi R^2$ to the elastic cross section. This is a purely wave-mechanical effect with no classical analogue.

17.9 The Optical Model

17.9.1 Motivation

In the 1950s, it was discovered that neutron total cross sections for medium and heavy nuclei exhibit a smooth, oscillating pattern as a function of energy (superimposed on sharp compound-nucleus resonances). These size resonances (also called shape resonances) correspond to constructive interference between waves reflected from the near and far sides of the nuclear potential — much like light passing through a glass sphere.

This observation led Feshbach, Porter, and Weisskopf (1954) to propose the optical model: treat the nucleus as a sphere described by a complex potential:

$$V(r) = V_R(r) + iW(r)$$

where $V_R(r)$ is the real (refractive) part and $W(r)$ is the imaginary (absorptive) part. The real part describes the mean-field potential that an incoming nucleon feels — it causes elastic scattering. The imaginary part describes the removal of flux from the elastic channel into all the inelastic and reaction channels — it causes absorption.

The analogy with optics is precise: a glass sphere has a complex index of refraction $n = n_R + in_I$, where the real part causes refraction and the imaginary part causes absorption (a "cloudy crystal ball," as the original authors called it).

17.9.2 The Optical Potential

The standard form of the optical potential for nucleon-nucleus scattering is:

$$V_{\text{opt}}(r) = V_C(r) - V_0\,f(r, R_V, a_V) - iW_0\,f(r, R_W, a_W) - iW_D\,\frac{d}{dr}f(r, R_D, a_D) + V_{SO}\,\frac{1}{r}\frac{d}{dr}f(r, R_{SO}, a_{SO})\,\boldsymbol{\ell}\cdot\boldsymbol{s}$$

where: - $V_C(r)$ is the Coulomb potential (for protons) - $f(r, R, a) = [1 + \exp((r - R)/a)]^{-1}$ is the Woods-Saxon form factor - $V_0 \approx 40$--$50\,\text{MeV}$: real volume depth - $W_0$: imaginary volume term (important at high energies, $E > 50\,\text{MeV}$) - $W_D \approx 5$--$15\,\text{MeV}$: imaginary surface term (dominant at low energies) - $V_{SO} \approx 5$--$8\,\text{MeV}$: real spin-orbit term

The radii follow $R_i = r_i A^{1/3}$ with $r_i \approx 1.2$--$1.3\,\text{fm}$, and the diffusenesses are $a_i \approx 0.5$--$0.7\,\text{fm}$.

17.9.3 Physical Interpretation

Real potential $V_R(r)$: This is the mean field that a nucleon experiences as it approaches and enters the nucleus — the same potential responsible for shell structure (Chapter 6). For incoming nucleons, it causes refraction of the de Broglie wave, producing elastic scattering angular distributions with characteristic diffraction-like minima and maxima.

Imaginary potential $W(r)$: This accounts for the loss of flux from the elastic channel. When a nucleon enters the nucleus, it can collide with individual nucleons, exciting them out of their ground-state orbits. These inelastic processes remove the nucleon from the elastic channel. The imaginary potential is the mathematical device that accomplishes this removal.

At low energies ($E < 50\,\text{MeV}$), the Pauli exclusion principle blocks collisions in the nuclear interior (there are no available empty states below the Fermi energy), so absorption occurs mainly at the nuclear surface — hence the surface-peaked imaginary term $W_D \cdot df/dr$. At high energies, collisions can occur throughout the nuclear volume, and the volume term $W_0$ dominates.

17.9.4 Predictions of the Optical Model

1. Elastic angular distributions: The optical model produces angular distributions with a characteristic diffraction pattern — a forward peak followed by oscillatory minima and maxima, similar to Fraunhofer diffraction from an opaque disk. The angular positions of the minima are approximately:

$$\theta_n \approx \frac{(n + 1/2)\pi}{kR}$$

where $n = 1, 2, 3, \ldots$ This provides a direct measure of the nuclear radius $R$.

2. Total cross sections: The energy dependence of $\sigma_{\text{tot}}$ shows broad oscillations with period $\Delta E \approx \pi\hbar v / (2R)$, corresponding to new partial waves entering the nuclear potential as the energy increases.

3. Reaction cross sections: The model predicts $\sigma_{\text{rxn}} \approx \pi R^2$ at high energies, modified at low energies by Coulomb barrier effects and the detailed form of the absorptive potential.

4. Size resonances: When the de Broglie wavelength inside the nucleus satisfies the condition for constructive interference ($2K R \approx n\pi$, where $K$ is the internal wave number), the cross section shows broad peaks — the 3s, 3p, 4s, ... shape resonances observed in neutron total cross sections.

The optical model, with about 10 adjustable parameters fitted to elastic scattering data, describes neutron and proton scattering from most nuclei across a broad energy range ($1$--$200\,\text{MeV}$) with impressive fidelity. Global optical model parametrizations (such as the Koning-Delaroche potential, 2003) provide predictions even for nuclei where no scattering data exist.

17.9.5 The Dispersion Relation

The real and imaginary parts of the optical potential are not independent. They are connected by a dispersion relation (analogous to the Kramers-Kronig relation in optics):

$$V_R(E) = V_{HF}(E) + \frac{\mathcal{P}}{\pi}\int_{-\infty}^{\infty}\frac{W(E')}{E' - E}\,dE'$$

where $V_{HF}$ is the energy-independent Hartree-Fock potential and $\mathcal{P}$ denotes the Cauchy principal value. This relation, derived from causality (the scattered wave cannot arrive before the incident wave), provides a powerful constraint: if the imaginary potential is known as a function of energy, the energy dependence of the real potential is determined. Dispersive optical models that enforce this constraint (Mahaux and Sartor, 1991) achieve significantly better fits to data than conventional parametrizations, particularly near the Fermi energy where the imaginary potential passes through zero.

17.9.6 Connection to Nuclear Structure

The optical model is intimately connected to nuclear structure through the nuclear mean field. The real part of the optical potential for a nucleon at the Fermi energy ($E \approx 0$) is essentially the shell-model potential (Chapter 6). As the nucleon energy increases above the Fermi energy, the potential weakens (the dispersive correction becomes negative) and the imaginary part grows (more phase space for inelastic collisions). This smooth evolution from the bound-state shell model to the scattering optical model — through the same underlying mean field — is one of the most satisfying unifications in nuclear physics.

The mean free path of a nucleon inside nuclear matter provides a physical measure of the absorptive potential's strength. For a nucleon with kinetic energy $T$ inside the nucleus (total energy $E + V_0$), the mean free path is approximately:

$$\lambda = \frac{\hbar v_{\text{int}}}{W} = \frac{\hbar\sqrt{2(E + V_0)/m}}{W}$$

At $E = 10\,\text{MeV}$ with $V_0 = 47\,\text{MeV}$ and $W = 10\,\text{MeV}$: $v_{\text{int}} \approx 0.33c$ and $\lambda \approx 6.5\,\text{fm}$ — comparable to the nuclear radius. This confirms the picture of moderate absorption: the nucleon typically traverses the nucleus once or twice before being absorbed, consistent with the observed diffraction patterns.

17.9.7 Limitations and Extensions

The optical model treats the target nucleus as an inert potential — it does not describe the internal excitations of the nucleus. It cannot predict the detailed spectrum of inelastic scattering or the angular distributions of specific reaction products. For those, one needs the compound nucleus model (Chapter 18) or the direct reaction theory (Chapter 19). The optical model provides the entrance channel description: how the projectile wave is distorted by the nuclear potential before and after the reaction.

Important extensions include:

• Coupled-channels (CC) calculations: Explicitly couple the elastic channel to low-lying collective excitations (rotational bands, vibrational phonons). This is essential for deformed nuclei, where the optical model alone misses the strong coupling between elastic scattering and Coulomb excitation of the first $2^+$ state.

• Microscopic optical potentials: Calculate the optical potential from the fundamental nucleon-nucleon interaction using nuclear many-body theory (the "folding model" or the nuclear matter approach). These parameter-free predictions test our understanding of the nuclear force in the medium.

• Exotic nuclei: For nuclei far from stability — halo nuclei with extended neutron distributions, for instance — standard global parametrizations fail. Dedicated optical potentials must be developed, often constrained by elastic scattering measurements at radioactive beam facilities.

17.10 Ericson Fluctuations

17.10.1 The Regime of Overlapping Resonances

At low beam energies, the cross section for neutron-induced reactions on medium and heavy nuclei shows isolated, well-separated resonances — individual excited states of the compound nucleus that can be resolved experimentally. As the excitation energy increases, the density of nuclear levels grows exponentially (Chapter 18 will quantify this), and eventually the resonances overlap. In this regime, the cross section as a function of energy becomes a rapidly and apparently randomly fluctuating function.

Torleif Ericson (1960) showed that these fluctuations are not noise but carry systematic statistical information about the compound nucleus. The key predictions:

17.10.2 Statistical Properties

1. Autocorrelation function: The energy autocorrelation function of the cross section fluctuations:

$$C(\epsilon) = \frac{\langle \sigma(E)\,\sigma(E + \epsilon)\rangle - \langle\sigma\rangle^2}{\langle\sigma\rangle^2}$$

has a Lorentzian form:

$$C(\epsilon) = \frac{1}{N_{\text{eff}}} \cdot \frac{\Gamma^2}{\epsilon^2 + \Gamma^2}$$

where $\Gamma$ is the average total width of the overlapping resonances and $N_{\text{eff}}$ is the effective number of contributing channels.

2. Correlation width: The half-width of the autocorrelation function directly gives the average resonance width $\Gamma$. This is experimentally accessible even when individual resonances cannot be resolved.

3. Cross-channel correlations: Fluctuations in different exit channels (e.g., $(n, n')$ and $(n, \alpha)$) are uncorrelated — each channel samples a different set of partial widths, and in the statistical (compound-nucleus) limit these are independent random variables.

4. Amplitude distribution: The cross section at a given energy, summed over many overlapping resonances with random phases, follows a $\chi^2$ distribution. For a single spin channel, the elastic cross section fluctuations follow a Porter-Thomas distribution ($\chi^2$ with 1 degree of freedom).

17.10.3 Experimental Observation

Ericson fluctuations have been observed in numerous systems, most cleanly in reactions on medium-mass nuclei at excitation energies of 15--25 MeV. Classic examples include:

• ${}^{24}\text{Mg}(p, p')$ at $E_p = 10$--$20\,\text{MeV}$: well-studied fluctuations with $\Gamma \approx 40\,\text{keV}$
• ${}^{60}\text{Ni}(p, \alpha)$ at $E_p = 8$--$18\,\text{MeV}$: fluctuations used to extract level densities

The analysis of Ericson fluctuations provides one of the few experimental handles on the properties of highly excited nuclear states — a regime where individual-level spectroscopy is impossible.

Connection to quantum chaos: Ericson fluctuations are now understood as a manifestation of quantum chaos in the compound nucleus. The statistical properties of the fluctuations (Lorentzian autocorrelation, Porter-Thomas width distribution) follow from random-matrix theory applied to the nuclear Hamiltonian — a topic we will explore further in Chapter 18.

17.11 Summary and Looking Ahead

This chapter has built the quantitative framework for nuclear reaction physics:

1. Reaction notation $a(b,c)d$ and the conservation laws (energy, momentum, charge, baryon number, lepton number, angular momentum, parity) that constrain what reactions are possible.

2. Q-values $Q = (M_{\text{initial}} - M_{\text{final}})c^2$ determine whether a reaction is exothermic ($Q > 0$) or endothermic ($Q < 0$), and the threshold energy $T_{\text{th}} = -Q(M_a + M_b + M_c + M_d)/(2M_a)$ is the minimum lab energy for endothermic reactions.

3. Center-of-mass kinematics — the CM energy $T_{\text{CM}} = T_{\text{lab}} M_a/(M_a + M_b)$, the lab-to-CM angle transformation, and the invariant mass $\sqrt{s}$ — are essential for connecting theory (CM frame) to experiment (lab frame).

4. Cross sections — differential ($d\sigma/d\Omega$) and total ($\sigma$) — are the fundamental observables, measured in barns, and are not geometric areas but quantum-mechanical probabilities.

5. The Rutherford scattering formula $d\sigma/d\Omega = (a/2)^2 / \sin^4(\Theta/2)$ was derived from the classical Coulomb orbit and confirmed to be the exact quantum result for distinguishable spinless charges.

6. Partial wave analysis decomposes the scattering into angular momentum components $l$, with phase shifts $\delta_l$ and transmission coefficients $T_l$ connecting the nuclear potential to observables.

7. The optical model $V(r) = V_R(r) + iW(r)$ treats the nucleus as a complex potential: the real part describes elastic scattering, the imaginary part describes absorption. It predicts diffraction patterns, shape resonances, and total cross sections across broad energy ranges.

8. Ericson fluctuations arise when compound-nucleus resonances overlap, producing apparently random cross section variations whose statistical properties encode the average level width $\Gamma$.

In Chapter 18, we will develop the compound nucleus model — Bohr's picture of a long-lived intermediate state that "forgets" how it was formed. The Breit-Wigner resonance formula will emerge as the natural description of isolated compound-nucleus resonances. In Chapter 19, we turn to direct reactions — fast, peripheral processes (stripping, pickup, knockout) that probe single-particle structure. Both models use the optical model wave functions developed here as their starting point.

The progressive project continues: reaction_kinematics.py (see code directory) implements Q-value calculations, threshold energies, and CM-to-lab transformations for any two-body reaction. You will extend this toolkit in Chapter 18 with Breit-Wigner cross sections and in Chapter 21 with fusion reaction rates.

References and Data Sources

The atomic mass data used throughout this chapter come from:

• Wang, M. et al., "The AME 2020 atomic mass evaluation (II)," Chinese Physics C 45, 030003 (2021). Available at: https://www-nds.iaea.org/amdc/

The optical model discussion draws on:

• Koning, A.J. and Delaroche, J.P., "Local and global nucleon optical models from 1 keV to 200 MeV," Nuclear Physics A 713, 231 (2003).
• Feshbach, H., Porter, C.E., and Weisskopf, V.F., "Model for Nuclear Reactions with Neutrons," Physical Review 96, 448 (1954).

The Ericson fluctuation theory:

• Ericson, T., "Fluctuations of Nuclear Cross Sections in the 'Continuum' Region," Physical Review Letters 5, 430 (1960).

The partial wave and scattering theory treatment follows:

• Krane, K.S., Introductory Nuclear Physics, Wiley (1987), Chapter 11.
• Wong, S.S.M., Introductory Nuclear Physics, 2nd ed., Wiley-VCH (2004), Chapter 8.
• Bertulani, C.A., Nuclear Physics in a Nutshell, Princeton (2007), Chapters 4 and 10.