Chapter 19 — Direct Reactions: Stripping, Pickup, and Knockout

"The direct reaction is the nuclear physicist's scalpel. Where the compound nucleus reaction probes the statistical properties of highly excited states, the direct reaction reaches in and touches individual nucleons." — Norman K. Glendenning, Direct Nuclear Reactions (1983)

Chapter Overview

In Chapter 18, we studied compound nucleus reactions — processes in which the projectile is absorbed, its energy shared among all nucleons, and the resulting excited system decays statistically, having "forgotten" how it was formed. Those reactions tell us about the average, statistical properties of nuclei at high excitation.

This chapter is about the opposite extreme. Direct reactions are fast — they happen in roughly the time it takes the projectile to traverse the nuclear diameter, about $10^{-22}$ s. They involve only the surface nucleons, not the entire nucleus. And far from forgetting the entrance channel, direct reactions carry detailed information about the quantum state of the transferred nucleon into the angular distribution of the outgoing particle.

This makes direct reactions the most powerful experimental probes of nuclear structure available to the experimentalist. A single (d,p) measurement can identify the orbital angular momentum, the total spin, and the occupation probability of a specific shell-model orbit. In the decades since the pioneering work of Butler (1951), direct reactions have provided the majority of our quantitative knowledge about nuclear single-particle structure — the experimental foundation upon which the shell model (Chapter 6) rests.

We will develop the subject in four stages:

• Section 19.1 introduces the characteristics that define direct reactions and distinguish them from compound nucleus processes.
• Section 19.2 develops the key reaction types: stripping, pickup, and knockout.
• Section 19.3 builds the theoretical framework — the distorted-wave Born approximation (DWBA) — that connects measured angular distributions to nuclear structure information.
• Section 19.4 introduces spectroscopic factors and the quenching problem, one of the most actively debated issues in modern nuclear structure physics.
• Section 19.5 describes how direct reactions are being extended to radioactive beam facilities, where inverse kinematics enables the study of exotic nuclei far from stability.

🏃 Fast Track: If you are primarily interested in the experimental technique, focus on Sections 19.1–19.2 and the spectroscopic factor discussion in 19.4. If you are interested in the theoretical formalism, the DWBA derivation in Section 19.3 is essential.

🔬 Deep Dive: The DWBA formalism connects directly to the scattering theory of Chapter 17 and the shell model of Chapter 6. Understanding Section 19.3 in depth prepares you for the modern ab initio structure calculations discussed in Chapter 31.

📊 Spaced Review (Chapter 17): Recall the cross section concept: $d\sigma/d\Omega$ measures the probability of scattering into solid angle $d\Omega$, and the optical model describes the nucleus as a complex potential that both refracts and absorbs the incoming wave. Both concepts are essential prerequisites for the DWBA.

📊 Spaced Review (Chapter 6): The shell model predicts that nucleons occupy discrete orbits labeled by quantum numbers $n$, $l$, $j$, with energies set by the mean-field potential plus the spin-orbit interaction. Direct reactions provide the most direct experimental test of these predictions.

19.1 What Makes a Reaction "Direct"?

19.1.1 Timescales: The Nuclear Clock

The most fundamental distinction between compound nucleus and direct reactions is time. Consider a projectile with kinetic energy $T \sim 10$–$50$ MeV incident on a medium-mass target ($A \sim 50$–$200$). The projectile's speed in the center-of-mass frame is:

$$v \sim \sqrt{\frac{2T}{m_N}} \sim 0.1c \text{ to } 0.3c$$

The time to traverse the nuclear diameter ($2R \approx 2 r_0 A^{1/3} \approx 10$–$14$ fm for medium to heavy nuclei) is:

$$\tau_{\text{transit}} = \frac{2R}{v} \sim \frac{12 \text{ fm}}{0.15c} \sim \frac{12 \text{ fm}}{4.5 \times 10^{22} \text{ fm/s}} \sim 3 \times 10^{-23} \text{ s}$$

This is the transit time — the natural timescale for a direct reaction. It is far shorter than the compound nucleus lifetime. Recall from Chapter 18 that the compound nucleus lives for $\tau_{\text{CN}} \sim \hbar/\Gamma \sim 10^{-19}$–$10^{-16}$ s for typical resonance widths $\Gamma \sim 1$ eV to $1$ keV. The ratio is:

$$\frac{\tau_{\text{CN}}}{\tau_{\text{transit}}} \sim 10^{4} \text{ to } 10^{7}$$

The compound nucleus lives thousands to millions of transit times — long enough for the energy to be shared statistically among all nucleons. A direct reaction, by contrast, is over before the rest of the nucleus has time to respond. This is the fundamental reason why direct reactions probe individual nucleons rather than the nucleus as a whole.

19.1.2 Experimental Signatures

How does the experimentalist distinguish a direct reaction from a compound nucleus reaction? Four signatures, listed from most to least diagnostic:

1. Angular distributions are forward-peaked and structured.

This is the defining experimental signature. Direct reaction angular distributions show a characteristic pattern of oscillations — a forward peak followed by minima and secondary maxima — with the number and positions of the minima determined by the angular momentum transferred. Compound nucleus angular distributions, by contrast, are symmetric about $90°$ in the center-of-mass frame (or nearly so), reflecting the compound nucleus's loss of memory of the entrance channel.

2. Energy dependence is smooth.

Direct reaction cross sections vary smoothly with bombarding energy, because they involve transitions between well-defined initial and final states. Compound nucleus cross sections are dominated by sharp resonance structure (Breit-Wigner peaks, Chapter 18) at lower energies, and by statistical fluctuations (Ericson fluctuations) at higher energies where resonances overlap.

3. Cross sections are sensitive to the specific final state.

A (d,p) reaction populating different states in the residual nucleus gives different angular distributions — the pattern depends on the quantum numbers of the final state. In a compound nucleus reaction, the decay is statistical: the branching to different final states depends on level densities and transmission coefficients, not on the specific structural overlap between initial and final states.

4. Reaction products are concentrated at forward angles.

Direct reactions are peripheral — they occur when the projectile grazes the nuclear surface. The transferred particle carries orbital angular momentum, and the matching between the momentum of the projectile and the momentum of the transferred nucleon in its bound orbit produces a forward-focused angular distribution. The grazing angular momentum is:

$$l_{\text{graze}} \approx k R$$

where $k$ is the projectile's wave number and $R$ is the sum of the nuclear radii. For 20 MeV deuterons on a medium-mass target, $l_{\text{graze}} \sim 5$–$8$, confirming the peripheral character.

19.1.3 The Continuum Between Direct and Compound

Real reactions do not fall neatly into "direct" or "compound" categories. Pre-equilibrium emission (also called pre-compound emission), in which one or a few nucleon-nucleon collisions occur before full equilibration, fills the gap. The energy spectrum of emitted particles in a nuclear reaction typically shows:

• A high-energy peak at the beam velocity — direct reactions (single-step transfer or knockout)
• A broad continuum at intermediate energies — pre-equilibrium emission
• A low-energy evaporation peak — compound nucleus decay (statistical, Maxwellian)

The relative importance of these components depends on bombarding energy. Below about 10 MeV per nucleon, compound nucleus reactions dominate. Above about 50 MeV per nucleon, direct reactions dominate. In between, all mechanisms coexist.

For this chapter, we focus on the cleanly direct regime, where the theoretical framework (DWBA) is most powerful and the connection to nuclear structure most transparent.

19.2 Types of Direct Reactions

Direct reactions come in several varieties, classified by what happens to the nucleons during the collision. We treat the three most important types in order of historical and practical significance.

19.2.1 Stripping Reactions: (d,p) and the Determination of l-Values

The most important direct reaction in nuclear physics is the deuteron stripping reaction, in which a deuteron ($d = {}^2\text{H}$, a loosely bound proton-neutron pair with binding energy $B_d = 2.225$ MeV) strikes a target nucleus $A$ and the neutron is stripped off and captured by the target, while the proton continues forward:

$$d + A \to p + (A+1)^*$$

The notation is $A(d,p)A+1$. The residual nucleus $(A+1)$ may be left in its ground state or in an excited state; by measuring the proton energy and angle, the experimentalist determines which final state was populated.

Why the deuteron? The deuteron is the ideal projectile for stripping reactions because:

1. It is loosely bound. The neutron and proton in the deuteron are separated by an average distance of about 4 fm — larger than the deuteron's own "nuclear radius." This means that as the deuteron grazes the target, the neutron can be close enough to be captured by the nuclear potential while the proton is still outside the range of the nuclear force. The reaction is naturally peripheral.

2. The internal structure is simple. The deuteron ground state is predominantly $^3S_1$ (with a small $^3D_1$ admixture), so the relative $n$-$p$ wavefunction is well known. This simplifies the theoretical analysis enormously.

3. The transferred particle carries well-defined quantum numbers. The neutron is transferred into a specific single-particle orbit in the residual nucleus. If it occupies an orbit with orbital angular momentum $l$, the angular distribution of the outgoing proton shows a characteristic pattern with $l$ minima in the first few oscillations.

Butler's Breakthrough (1951)

The theoretical understanding of (d,p) reactions was revolutionized by S.T. Butler's 1951 paper, which showed that the angular distribution of the outgoing proton is controlled by the orbital angular momentum $l$ of the captured neutron. In the simplest approximation — treating the incoming deuteron and outgoing proton as plane waves (the plane-wave Born approximation, PWBA) — Butler showed that the transition amplitude is proportional to the Fourier transform of the neutron's bound-state wavefunction:

$$T_{\text{PWBA}} \propto \int \phi_{\text{bound}}(\mathbf{r}) \, e^{i\mathbf{q}\cdot\mathbf{r}} \, d^3r$$

where $\mathbf{q} = \mathbf{k}_d - \mathbf{k}_p$ is the momentum transfer. For a neutron bound in an orbit with orbital angular momentum $l$, the bound-state wavefunction contains a spherical harmonic $Y_{lm}$, and the Fourier transform produces a spherical Bessel function $j_l(qR)$. The angular distribution therefore has the oscillatory structure of $|j_l(qR)|^2$:

• $l = 0$: $j_0(x) = \sin(x)/x$ — a single forward peak with no minimum at $0°$.
• $l = 1$: $j_1(x) = \sin(x)/x^2 - \cos(x)/x$ — one minimum in the forward hemisphere.
• $l = 2$: $j_2(x)$ — two minima.
• $l = 3$: $j_3(x)$ — three minima.

The general rule: the number of minima in the angular distribution (before the pattern washes out) equals the transferred orbital angular momentum $l$. This is the most important result in the chapter, and it is the reason that (d,p) reactions are the workhorse tool for determining nuclear structure.

Example: ${}^{208}\text{Pb}(d,p){}^{209}\text{Pb}$

The classic application is to the doubly magic nucleus ${}^{208}\text{Pb}$. When a deuteron strips its neutron onto ${}^{208}\text{Pb}$, the neutron must go into the first available orbit above the $N = 126$ shell closure. The shell model (Chapter 6) predicts the available orbits above $N = 126$ to be $2g_{9/2}$, $1i_{11/2}$, $1j_{15/2}$, $3d_{5/2}$, $4s_{1/2}$, $2g_{7/2}$, $3d_{3/2}$.

The angular distributions measured for ${}^{208}\text{Pb}(d,p){}^{209}\text{Pb}$ at $E_d = 20$ MeV by Ellegaard et al. (1971) show:

Let us be precise with the known spectroscopy. The measured states of ${}^{209}\text{Pb}$ and their (d,p) assignments are:

The $l = 0$ transfer to the $4s_{1/2}$ state produces a featureless forward peak. The $l = 2$ transfers to the $3d$ states show two clear minima. The $l = 4$ transfer to the $2g_{9/2}$ ground state shows four minima (the first two are clearly resolved, the higher-order ones are damped). Each angular distribution is a fingerprint of the orbit, and the complete set maps out the single-particle spectrum above the $N = 126$ shell closure — confirming the shell model predictions with remarkable precision.

💡 Key Insight: The (d,p) reaction on a doubly magic target is the cleanest possible probe of single-particle structure, because the transferred neutron goes into a single well-defined orbit. This is why ${}^{208}$Pb, ${}^{48}$Ca, ${}^{16}$O, and ${}^{40}$Ca are the benchmark nuclei for testing the shell model.

19.2.2 Pickup Reactions: (p,d) and Hole States

The time-reverse of stripping is pickup: a proton incident on a target picks up a neutron from the target to form a deuteron:

$$p + A \to d + (A-1)^*$$

The notation is $A(p,d)A-1$. Now the residual nucleus has one fewer neutron than the target, and the angular distribution of the outgoing deuteron reveals the $l$-value of the orbit from which the neutron was removed.

Pickup reactions probe hole states — states created by removing a nucleon from a filled orbit. If the target has a closed neutron shell, the pickup reaction reveals the single-particle energies of the orbits below the Fermi surface, just as stripping reveals the orbits above it. Together, stripping and pickup provide a complete map of the single-particle spectrum around the Fermi energy — the fundamental input for the shell model.

Example: ${}^{208}\text{Pb}(p,d){}^{207}\text{Pb}$

Removing a neutron from ${}^{208}\text{Pb}$ creates states in ${}^{207}\text{Pb}$. The shell model predicts that the neutron orbits just below $N = 126$ are $3p_{1/2}$, $2f_{5/2}$, $3p_{3/2}$, $1i_{13/2}$, $2f_{7/2}$, $1h_{9/2}$ (filling the $N = 82$–$126$ shell). The measured single-hole states and their pickup angular distributions confirm these assignments:

Together, the (d,p) and (p,d) data on ${}^{208}$Pb provide a complete picture of the single-particle spectrum spanning roughly 10 MeV on either side of the Fermi surface. This is, in a real sense, the experimental verification of the shell model.

19.2.3 Other Transfer Reactions

The (d,p) and (p,d) reactions are the most common, but they are members of a large family of single-nucleon transfer reactions:

• $A({}^3\text{He},d)B$ — proton stripping (analogous to neutron stripping with (d,p))
• $A(d,{}^3\text{He})B$ — proton pickup
• $A(t,d)B$ — neutron stripping using tritons
• $A(\alpha,{}^3\text{He})B$ — neutron stripping using alpha particles

Two-nucleon transfer reactions such as $(t,p)$ and $(p,t)$ transfer a pair of neutrons and probe pairing correlations — the tendency of nucleons to form $J = 0$ pairs in the nuclear ground state. These are important tests of the pairing interaction in the nuclear Hamiltonian but lie beyond our scope here.

19.2.4 Knockout Reactions: (e,e'p) and (p,2p)

A qualitatively different type of direct reaction is the knockout reaction, in which a single nucleon is ejected from the target by a probe that interacts with it directly. The most important knockout reactions are:

Electron-induced knockout: $(e,e'p)$

An electron scatters from a proton inside the nucleus, ejecting it. By measuring the scattered electron and the knocked-out proton in coincidence, the experimentalist determines:

• The missing energy: $E_{\text{miss}} = \omega - T_p - T_{A-1}$, where $\omega$ is the energy transferred by the electron, $T_p$ is the proton kinetic energy, and $T_{A-1}$ is the recoil kinetic energy of the residual nucleus. This identifies the separation energy of the removed proton.

• The missing momentum: $\mathbf{p}_{\text{miss}} = \mathbf{q} - \mathbf{p}_p$, where $\mathbf{q}$ is the momentum transferred by the electron and $\mathbf{p}_p$ is the outgoing proton momentum. This is equal and opposite to the initial momentum of the proton inside the nucleus.

The missing-momentum distribution is directly related to the momentum-space wavefunction of the knocked-out proton in its shell-model orbit: $|\tilde{\phi}_{nlj}(\mathbf{p}_{\text{miss}})|^2$. Different orbits produce characteristic distributions — an $s$-state has maximum amplitude at $p_{\text{miss}} = 0$, a $p$-state has a node at zero, and higher-$l$ orbits show correspondingly more nodes.

The great advantage of $(e,e'p)$ over transfer reactions is that the electromagnetic interaction is well understood (QED), so the reaction mechanism introduces minimal theoretical uncertainty. The electron does not interact via the strong force, so it can probe deeply into the nuclear interior, not just the surface.

Landmark result: NIKHEF ${}^{208}\text{Pb}(e,e'p)$ measurements

The $(e,e'p)$ measurements performed at NIKHEF (Amsterdam) in the 1990s and early 2000s, particularly those of Quint et al. (1988) and subsequent analyses, provided the most precise determinations of proton spectroscopic factors in stable nuclei. For the deeply bound proton orbits in ${}^{208}\text{Pb}$, they found spectroscopic factors of $0.6$–$0.7$ — significantly less than the shell-model prediction of 1.0. This "quenching" of the spectroscopic factor (Section 19.4) is one of the most important open questions in nuclear structure physics.

Proton-induced knockout: $(p,2p)$ and $(p,pn)$

At radioactive beam facilities, where the exotic nucleus is the beam particle, electron scattering is not possible (you cannot make an electron target). Instead, quasi-free knockout reactions are performed using a proton target (typically a liquid hydrogen target or a CH$_2$ target):

• $(p,2p)$: knocks out a proton, leaving the $(A-1)$ residue
• $(p,pn)$: knocks out a neutron

These reactions are performed in inverse kinematics — the radioactive beam strikes the proton target — and have become the primary tool for studying single-particle structure in exotic nuclei at facilities like RIKEN (Japan), GSI/FAIR (Germany), and FRIB (USA). We return to this in Section 19.5.

19.3 The Distorted-Wave Born Approximation (DWBA)

19.3.1 Why We Need More Than Plane Waves

Butler's plane-wave theory captures the essential physics — the connection between $l$ and the angular distribution pattern — but it has serious quantitative limitations. The incoming deuteron and outgoing proton are not plane waves; they are strongly distorted by the nuclear potential. Ignoring this distortion gives:

1. Angular distributions that are too sharply peaked.
2. Cross section magnitudes that can be wrong by an order of magnitude.
3. Incorrect positions of the diffraction minima.

The remedy is the distorted-wave Born approximation (DWBA), developed in the late 1950s and early 1960s by Tobocman, Austern, Satchler, and others. The DWBA treats the nuclear reaction as a perturbation on top of elastic scattering, using the optical model wavefunctions (Chapter 17) for the entrance and exit channels.

19.3.2 Formal Development

Consider a stripping reaction $A(d,p)B$, where $B = A + n$. We denote the entrance channel as $\alpha = d + A$ and the exit channel as $\beta = p + B$. The full Hamiltonian is:

$$H = T_\alpha + V_\alpha + H_A + H_d$$

where $T_\alpha$ is the kinetic energy of relative motion in the entrance channel, $V_\alpha$ is the interaction between the deuteron and the target, $H_A$ is the internal Hamiltonian of the target, and $H_d$ is the internal Hamiltonian of the deuteron.

We can also write the same Hamiltonian in exit-channel coordinates:

$$H = T_\beta + V_\beta + H_B + H_p$$

The key trick of the DWBA is to split the interaction in each channel into an optical potential $U$ (which describes elastic scattering) and a residual interaction $\Delta V$ (which causes the transfer):

$$V_\alpha = U_\alpha + \Delta V_\alpha$$

The DWBA transition amplitude is then:

$$T_{\text{DWBA}} = \langle \chi^{(-)}_\beta \, \phi_B \, \phi_p \,|\, \Delta V \,|\, \chi^{(+)}_\alpha \, \phi_A \, \phi_d \rangle$$

where:

• $\chi^{(+)}_\alpha(\mathbf{R}_\alpha)$ is the distorted wave in the entrance channel — the solution of the Schrodinger equation with the optical potential $U_\alpha$, with outgoing-wave boundary conditions (+).
• $\chi^{(-)}_\beta(\mathbf{R}_\beta)$ is the distorted wave in the exit channel, with incoming-wave boundary conditions (−) (time-reversed scattering state).
• $\phi_d$, $\phi_A$, $\phi_B$, $\phi_p$ are the internal wavefunctions of the deuteron, target, residual nucleus, and proton.
• $\Delta V$ is the residual interaction that causes the transfer.

19.3.3 The Transfer Form Factor

The nuclear structure information enters through the overlap function (also called the form factor) between the target and residual nucleus:

$$\phi_B(\mathbf{r}_1, \ldots, \mathbf{r}_{A+1}) \approx \mathcal{A}\left[\phi_A(\mathbf{r}_1, \ldots, \mathbf{r}_A) \times \varphi_{nlj}(\mathbf{r}_n)\right]$$

where $\varphi_{nlj}(\mathbf{r}_n)$ is the single-particle wavefunction of the transferred neutron in the residual nucleus, with quantum numbers $n$, $l$, $j$. The antisymmetrization operator $\mathcal{A}$ ensures proper Fermi statistics. The overlap function is normalized such that:

$$\langle \phi_B | \phi_A \times \varphi_{nlj} \rangle = \sqrt{S_{nlj}}$$

where $S_{nlj}$ is the spectroscopic factor — the probability amplitude that the state $B$ can be described as nucleus $A$ plus a neutron in orbit $nlj$.

After integrating over the internal coordinates of the target and residual nucleus, the DWBA amplitude takes the form:

$$T_{\text{DWBA}} = \sqrt{S_{nlj}} \int \chi^{(-)}_\beta(\mathbf{R}_\beta)^* \, F_{nlj}(\mathbf{r}) \, \chi^{(+)}_\alpha(\mathbf{R}_\alpha) \, d^3R_\alpha \, d^3r$$

where $F_{nlj}(\mathbf{r})$ incorporates the deuteron wavefunction, the neutron bound-state wavefunction $\varphi_{nlj}(\mathbf{r})$, and the residual interaction $\Delta V$.

The differential cross section is:

$$\frac{d\sigma}{d\Omega} = \frac{\mu_\alpha \mu_\beta}{(2\pi\hbar^2)^2} \frac{k_\beta}{k_\alpha} \frac{1}{(2J_A + 1)(2J_d + 1)} \sum_{M_A, M_d, M_B, M_p} |T_{\text{DWBA}}|^2$$

After performing the spin sums and angular momentum algebra, this becomes:

$$\frac{d\sigma}{d\Omega} = S_{nlj} \cdot \sigma_{\text{DWBA}}^{sp}(\theta)$$

where $\sigma_{\text{DWBA}}^{sp}(\theta)$ is the single-particle cross section — the DWBA prediction assuming a pure single-particle state ($S = 1$). This factorization is the central result of DWBA theory: the measured cross section equals the spectroscopic factor times the theoretical single-particle cross section.

19.3.4 The Optical Potentials

The distorted waves $\chi^{(\pm)}$ are obtained by solving the Schrodinger equation with the optical potential:

$$\left[-\frac{\hbar^2}{2\mu}\nabla^2 + U(r) + V_C(r)\right]\chi(\mathbf{r}) = E\chi(\mathbf{r})$$

where $U(r)$ is the nuclear optical potential and $V_C(r)$ is the Coulomb potential. As discussed in Chapter 17, the optical potential has the general form:

$$U(r) = -V_0 \, f(r; R_V, a_V) - i W_V \, f(r; R_W, a_W) - i 4a_W' W_D \frac{d}{dr}f(r; R_W', a_W') + V_{so}\left(\frac{\hbar}{m_\pi c}\right)^2 \frac{1}{r}\frac{d}{dr}f(r; R_{so}, a_{so})\,\boldsymbol{\ell}\cdot\boldsymbol{s}$$

where $f(r; R, a) = [1 + \exp((r-R)/a)]^{-1}$ is the Woods-Saxon form factor. The parameters $V_0$, $W_V$, $W_D$, $V_{so}$ and the various radii and diffusenesses are determined by fitting elastic scattering data in the entrance and exit channels. This is a crucial point: the optical potentials are not free parameters in the DWBA analysis — they are fixed by independent elastic scattering measurements.

19.3.5 The Bound-State Wavefunction

The wavefunction of the transferred neutron in its bound orbit, $\varphi_{nlj}(\mathbf{r})$, is obtained by solving:

$$\left[-\frac{\hbar^2}{2\mu_n}\nabla^2 + V_{\text{WS}}(r) + V_{so}(r)\boldsymbol{\ell}\cdot\boldsymbol{s}\right]\varphi_{nlj}(\mathbf{r}) = -B_n \varphi_{nlj}(\mathbf{r})$$

where $V_{\text{WS}}(r)$ is a Woods-Saxon potential and $B_n$ is the neutron separation energy (known experimentally from the $Q$-value of the reaction). The radius and diffuseness of the binding potential are typically fixed at "standard" values ($r_0 = 1.25$ fm, $a = 0.65$ fm), and the depth $V_0$ is adjusted to reproduce the known separation energy. This procedure ensures that the asymptotic behavior of the bound-state wavefunction — which dominates the cross section for peripheral reactions — is correct.

19.3.6 Putting It All Together: A DWBA Calculation

A complete DWBA analysis of a (d,p) reaction therefore requires:

1. Optical potential in the entrance channel ($d + A$): fit to elastic scattering of deuterons from the target at the appropriate energy.
2. Optical potential in the exit channel ($p + B$): fit to elastic scattering of protons from the residual nucleus (or a nearby nucleus) at the appropriate energy.
3. Bound-state wavefunction: calculated in a Woods-Saxon potential with the depth adjusted to give the correct separation energy, and the quantum numbers ($n$, $l$, $j$) chosen for the orbit being tested.
4. Deuteron wavefunction: usually a Hulthen form or a numerical solution of the $n$-$p$ Schrodinger equation with a realistic $NN$ potential.
5. Numerical integration: the six-dimensional integral in the DWBA amplitude is evaluated numerically (typically after partial-wave decomposition, which reduces it to a sum of radial integrals).

The resulting angular distribution is compared to the data. The $l$-value is determined by the shape of the angular distribution; the spectroscopic factor is determined by its magnitude:

$$S_{nlj} = \frac{(d\sigma/d\Omega)_{\text{exp}}}{(d\sigma/d\Omega)_{\text{DWBA}}^{sp}} \bigg|_{\text{first maximum}}$$

In practice, the comparison is made over the full angular range, not just at the first maximum, using a least-squares fit.

Modern DWBA codes — DWUCK (Kunz), TWOFNR (Igarashi), and FRESCO (Thompson) — perform these calculations routinely. A DWBA analysis of a single (d,p) angular distribution takes seconds on a modern computer, and the extracted $l$-values are robust: they depend primarily on the shape of the distorted waves, which is well constrained by the optical potentials.

💡 Key Insight: The DWBA factorizes the reaction cross section into a nuclear structure part (the spectroscopic factor) and a reaction mechanism part (the single-particle DWBA cross section). This factorization is what makes direct reactions such powerful structure probes — the reaction mechanism is calculable, so the structure information can be extracted.

19.3.7 Worked Example: DWBA Analysis of ${}^{48}\text{Ca}(d,p){}^{49}\text{Ca}$

To illustrate the DWBA procedure concretely, consider the reaction ${}^{48}\text{Ca}(d,p){}^{49}\text{Ca}$ at $E_d = 13$ MeV — a classic case because ${}^{48}\text{Ca}$ is doubly magic ($Z = 20$, $N = 28$), so the transferred neutron enters a well-defined single-particle orbit above the $N = 28$ shell closure.

Step 1: Identify the final states. The shell model predicts the neutron orbits above $N = 28$ to be $2p_{3/2}$, $2p_{1/2}$, $1f_{5/2}$, $1g_{9/2}$. The measured spectrum of ${}^{49}\text{Ca}$ confirms:

Step 2: Optical potentials. Elastic scattering of 13 MeV deuterons from ${}^{48}\text{Ca}$ is fit with a global optical potential (e.g., An and Cai, 2006) yielding parameters $V_0 \approx 90$ MeV, $r_V = 1.17$ fm, $a_V = 0.75$ fm, $W_D \approx 12$ MeV for the entrance channel. The exit-channel proton optical potential at the appropriate energy ($E_p \approx 18$–$20$ MeV, depending on the final state) uses the Koning-Delaroche global parameterization.

Step 3: Bound-state wavefunction. For the ground-state transition ($2p_{3/2}$), the neutron is bound by $S_n = 5.146$ MeV. A Woods-Saxon potential with $r_0 = 1.25$ fm, $a = 0.65$ fm, $V_{so} = 6$ MeV is used, and the central depth is adjusted to $V_0 = 53.2$ MeV to reproduce this binding energy with one node in the radial wavefunction (the $2p$ orbit has $n = 2$, meaning one radial node).

Step 4: DWBA calculation. The code FRESCO (or DWUCK) computes the six-dimensional transfer integral after partial-wave decomposition. The resulting angular distribution for the $l = 1$ ground-state transition shows:

• A strong forward peak at $\theta_{\text{CM}} \approx 5°$–$10°$
• One clear minimum at $\theta_{\text{CM}} \approx 35°$
• A secondary maximum near $50°$
• A gradual falloff toward larger angles

The single minimum in the forward hemisphere is the diagnostic signature of $l = 1$. For comparison, the $l = 3$ transition to the $1f_{5/2}$ state at $3.585$ MeV shows three minima before the pattern is damped — the peak is shifted to larger angle ($\sim 20°$), and the oscillation period is shorter.

Step 5: Extract spectroscopic factor. Normalizing the DWBA calculation to the data at the first maximum gives $S(2p_{3/2}) = 0.89 \pm 0.12$, close to the shell-model prediction of $S = 1$ for adding a neutron to an empty orbit. The reduction from unity is consistent with the systematic quenching discussed in Section 19.4, arising from correlations beyond the mean field.

This example illustrates the complete workflow: measure the angular distribution, calculate the DWBA with independently determined inputs, identify $l$ from the shape, and extract $S$ from the magnitude. The same procedure, applied systematically across the nuclear chart, has produced the majority of our knowledge about single-particle structure.

19.3.8 Finite-Range and Coupled-Channel Effects

The "standard" DWBA described above makes two approximations that can break down:

Zero-range approximation. In the simplest DWBA, the deuteron is treated as a point particle — the transfer occurs at a single point in space. A more accurate treatment uses the finite-range DWBA, which explicitly includes the spatial extent of the deuteron wavefunction. This changes the angular distributions by 10–30% and is now standard in modern analyses.

Single-step approximation. The DWBA assumes that the transfer is a single-step process — the system goes directly from the entrance channel to the exit channel. In reality, multi-step processes (e.g., the deuteron first exciting the target, then transferring the neutron) can contribute. These are treated in the coupled-channels Born approximation (CCBA) or the full coupled-reaction-channels (CRC) method, implemented in codes like FRESCO. For strongly deformed nuclei, where the coupling between ground-state and rotational excitations is strong, CCBA can be essential.

19.4 Spectroscopic Factors and the Quenching Problem

19.4.1 Definition and Physical Meaning

The spectroscopic factor $S_{nlj}$ is defined as the norm of the overlap function between the $A$-body target state and the $(A+1)$-body residual state:

$$S_{nlj} = |\langle \Phi_B | a^\dagger_{nlj} | \Phi_A \rangle|^2$$

where $a^\dagger_{nlj}$ creates a nucleon in orbit $nlj$. Physically, $S_{nlj}$ measures the probability that the state $B$ can be described as a nucleon added to (or removed from) the state $A$ in a specific orbit. In the independent-particle shell model:

• For stripping onto a closed-shell target (adding a neutron to an empty orbit): $S = 1$.
• For pickup from a closed-shell target (removing a neutron from a full orbit): $S = 2j + 1$ (the number of nucleons in the filled orbit, if we use the convention where the single-particle cross section is calculated for a single nucleon).

The conventional normalization is $0 \leq S \leq 2j + 1$, with $S/(2j+1)$ representing the fractional occupation of the orbit. For simplicity, we often use the convention where $S$ is normalized per nucleon, giving $0 \leq S \leq 1$.

19.4.2 Sum Rules

The spectroscopic factors satisfy sum rules that follow from the anticommutation relations of the creation and annihilation operators. The most important is the Macfarlane-French sum rule:

$$\sum_f S_{nlj}^{(+)}(A \to A+1_f) - \sum_f S_{nlj}^{(-)}(A \to A-1_f) = \frac{(2j+1) - 2\langle n_{nlj} \rangle_A}{1}$$

where the first sum runs over all final states populated by stripping (adding a nucleon), the second over all states populated by pickup (removing a nucleon), and $\langle n_{nlj} \rangle_A$ is the average occupation number of orbit $nlj$ in the target ground state. For a closed-shell target like ${}^{208}$Pb, $\langle n_{nlj} \rangle = 2j+1$ for orbits below the Fermi surface and 0 above.

A simpler sum rule for a single type of reaction:

$$\sum_f S_{nlj}^{(+)} = (2j+1)(1 - \bar{n}_{nlj})$$

where $\bar{n}_{nlj}$ is the fractional occupation. For stripping onto a closed shell ($\bar{n} = 0$), the sum of all stripping spectroscopic factors to a given orbit should equal $2j + 1$.

19.4.3 The Quenching Problem

One of the most important and persistent results in nuclear structure physics is that measured spectroscopic factors are systematically smaller than shell-model predictions. This was first seen in the $(e,e'p)$ measurements at NIKHEF and Saclay in the 1980s and 1990s, and it has been confirmed across a wide range of nuclei and reaction probes.

The typical finding: for well-bound orbits in closed-shell nuclei, the ratio of the experimental spectroscopic factor to the independent-particle model (IPM) prediction is:

$$R_s = \frac{S_{\text{exp}}}{S_{\text{IPM}}} \approx 0.55 \text{--} 0.70$$

This means that the occupancy of single-particle orbits is only 55–70% of what the naive shell model predicts. The "missing" strength is distributed over a wide range of excitation energies, spread by correlations that are not captured in the mean-field picture.

Where does the missing strength go? Several mechanisms contribute:

1. Short-range correlations (SRC). The strong repulsive core of the nucleon-nucleon interaction at distances less than about 0.5 fm causes high-momentum components in the nuclear wavefunction. These deplete the occupation of low-lying orbits and populate high-momentum, high-energy states. Measurements at Jefferson Lab using $(e,e'pp)$ and $(e,e'pn)$ reactions have shown that about 20% of nucleons are in correlated pairs at any given time, predominantly $np$ pairs.

2. Long-range correlations (LRC). Collective excitations — surface vibrations, giant resonances — couple to the single-particle motion and fragment the single-particle strength over many states. The random phase approximation (RPA) and its extensions quantify this effect.

3. Tensor correlations. The tensor component of the nuclear force (Chapter 3) causes strong $np$ correlations that deplete $l = 0$ orbits more than higher-$l$ orbits.

Modern nuclear structure theory — including self-consistent Green's function methods, coupled-cluster theory, and the dispersive optical model (DOM) — can now compute spectroscopic factors from first principles. The agreement with experiment is improving but remains an active area of research.

⚠️ Caution: Spectroscopic factors extracted from transfer reactions carry a systematic uncertainty of 20–30% due to the dependence on the choice of optical potentials, the bound-state geometry, and higher-order corrections (multi-step processes). Spectroscopic factors from $(e,e'p)$ reactions are considered more reliable, but they are limited to stable nuclei. Comparing spectroscopic factors extracted from different probes and by different analysis groups requires careful attention to the conventions and inputs used.

19.4.4 The Asymmetry Dependence

A remarkable result from radioactive beam experiments is that the quenching of spectroscopic factors depends on the neutron-proton asymmetry $(N-Z)/A$ of the nucleus. Measurements at NSCL (Michigan State) by Gade et al. (2004, 2008) using $(p,2p)$ and $(^9\text{Be}, X)$ knockout reactions on exotic nuclei showed that:

• For nuclei near the valley of stability, the quenching is about 60–70% of the IPM value ($R_s \approx 0.6$–$0.7$).
• For removal of a deeply bound minority species (e.g., removing a proton from a very neutron-rich nucleus), the quenching is much stronger: $R_s$ can be as low as $0.3$–$0.4$.
• For removal of a loosely bound majority species (e.g., removing a neutron from a neutron-rich nucleus), the quenching is weaker: $R_s \approx 0.8$–$0.9$.

This asymmetry dependence remains incompletely understood and is an active research topic. It may be related to the different isospin character of long-range and short-range correlations, or it may reflect limitations in the reaction theory used to extract spectroscopic factors from knockout experiments. Resolving this question is one of the scientific motivations for the Facility for Rare Isotope Beams (FRIB).

19.5 Direct Reactions with Radioactive Beams

19.5.1 The Need for Inverse Kinematics

The classic direct reaction measurements described above all used stable beams and stable targets — deuterium gas targets or thin foils of ${}^{208}$Pb, ${}^{48}$Ca, etc. But the most exciting frontier of nuclear structure lies far from stability: in the neutron-rich nuclei produced in supernovae and neutron star mergers, in the proton-rich nuclei relevant to X-ray bursts, in the exotic nuclei near the drip lines where new phenomena emerge (halos, shell evolution, new magic numbers).

These nuclei cannot be used as targets — they are radioactive and decay in milliseconds to seconds. Instead, they must be produced as beams and directed onto stable targets. This reversal of the roles of beam and target is called inverse kinematics.

In inverse kinematics, a radioactive beam $A$ strikes a light stable target (typically hydrogen, deuterium, or ${}^{9}\text{Be}$), and the reaction products are detected in the forward direction. The kinematics are very different from the normal case:

• The center-of-mass frame moves rapidly forward in the lab, so all products are focused into a forward cone.
• The energy and angle of the light ejectile (proton or deuteron) are measured, often with an array of silicon detectors surrounding the target.
• The heavy residue is identified downstream by its charge $Z$, mass $A$, and momentum, using magnetic spectrometers.
• The excitation energy of the residual nucleus is reconstructed from the measured momenta.

19.5.2 Transfer Reactions in Inverse Kinematics: (d,p) with Radioactive Beams

The $(d,p)$ reaction in inverse kinematics works as follows: a radioactive beam of nucleus $A$ at $E/A \sim 5$–$30$ MeV/nucleon bombards a deuterium target (CD$_2$ or liquid D$_2$). The neutron is stripped from the deuteron and added to the beam particle, forming $A+1$. The proton is detected at backward lab angles (because in the CM frame it goes forward, but the CM frame is boosted).

The silicon array ORRUBA (Oak Ridge Rutgers University Barrel Array), the SHARC (Silicon Highly-segmented Array for Reactions and Coulex) at TRIUMF/ISAC, and the HiRA (High Resolution Array) at NSCL/FRIB are designed for exactly these measurements.

Example: ${}^{132}\text{Sn}(d,p){}^{133}\text{Sn}$

${}^{132}\text{Sn}$ ($Z = 50$, $N = 82$) is a doubly magic nucleus far from stability — the neutron-rich analog of ${}^{208}$Pb. A landmark experiment at ORNL (Jones et al., 2010, published in Nature) measured the $(d,p)$ reaction on a ${}^{132}\text{Sn}$ beam at 4.8 MeV/nucleon, in inverse kinematics. The angular distributions of the outgoing protons revealed:

• The $2f_{7/2}$ neutron orbit above $N = 82$ at $E_x = 0$ (ground state of ${}^{133}\text{Sn}$), with $l = 3$.
• The $3p_{3/2}$ orbit at $E_x = 0.854$ MeV, with $l = 1$.
• The $3p_{1/2}$ orbit at $E_x = 1.363$ MeV, with $l = 1$.
• The $1h_{9/2}$ orbit at $E_x = 1.561$ MeV, with $l = 5$.
• The $2f_{5/2}$ orbit at $E_x = 2.005$ MeV, with $l = 3$.

These measurements provided the first direct determination of single-particle energies above $N = 82$ in the ${}^{132}\text{Sn}$ region — crucial data for understanding shell evolution far from stability and for $r$-process nucleosynthesis models (Chapter 23). The measured single-particle spacings differ significantly from those in ${}^{208}\text{Pb}$, demonstrating that the single-particle spectrum evolves with neutron-proton asymmetry — a key prediction of modern shell-model theory.

19.5.3 Knockout Reactions at Intermediate Energies

At higher beam energies ($E/A \gtrsim 50$ MeV/nucleon), one-nucleon knockout on light targets (${}^{9}\text{Be}$, ${}^{12}\text{C}$) becomes a powerful spectroscopic tool. The idea is simple: the beam particle grazes the target, and a single nucleon is knocked out (absorbed by the target). The residual $(A-1)$ fragment continues forward and is identified by its mass, charge, and longitudinal momentum distribution.

The cross section is:

$$\sigma_{\text{knockout}} = \sum_{nlj} S_{nlj} \cdot \sigma_{sp}^{nlj}$$

where $\sigma_{sp}^{nlj}$ is the single-particle knockout cross section (calculable from eikonal reaction theory) and $S_{nlj}$ is the spectroscopic factor. The momentum distribution of the residual fragment reflects the momentum distribution of the knocked-out nucleon in its orbit — an $s$-state gives a broad distribution (high momentum components from being spatially localized), while higher-$l$ states give narrower distributions.

Gamma-ray spectroscopy in coincidence with the knockout residue identifies the final state of the $(A-1)$ nucleus, enabling state-by-state spectroscopic factor extraction.

Major experimental programs at RIKEN (Radioactive Isotope Beam Factory), GSI/FAIR, NSCL (now FRIB), and GANIL have used this technique to study the single-particle structure of hundreds of exotic nuclei, from light $p$-shell nuclei to the medium-mass region around ${}^{68}\text{Ni}$ and ${}^{78}\text{Ni}$ to the neutron-rich oxygen and carbon isotopes where new magic numbers emerge.

FRIB and the next frontier. The Facility for Rare Isotope Beams (FRIB) at Michigan State University, which began user operations in 2022, represents a qualitative leap in rare-isotope capability. Its 400 kW superconducting linear accelerator can produce beams of over 1,000 isotopes that have never been studied before — many of them on or near the neutron drip line. The knockout reaction program at FRIB focuses on several key questions:

• Shell structure at $N = 50$ far from stability. Knockout reactions on exotic isotopes like ${}^{78}\text{Ni}$ ($Z = 28$, $N = 50$, doubly magic far from stability) will test whether the magic numbers persist or erode in extremely neutron-rich systems. The first spectroscopy of states in ${}^{79}\text{Cu}$ via the ${}^{79}\text{Cu}(p,2p){}^{78}\text{Ni}$ reaction is a flagship FRIB experiment.

• Mapping the $r$-process path. Nuclei on the rapid neutron-capture ($r$-process) path in neutron star mergers are extraordinarily neutron-rich. Their single-particle structure — accessible through knockout and transfer reactions at FRIB — determines the neutron separation energies and shell gaps that shape the $r$-process abundance pattern (Chapter 23). Every spectroscopic factor measured for an exotic nucleus near the $r$-process path is a direct input to nucleosynthesis models.

• Correlations and short-range physics. The $(p,2p)$ and $(p,pn)$ reactions at FRIB, performed at beam energies of $\sim 200$ MeV/nucleon with the GRETINA/GRETA gamma-ray tracking array and the HRS (High Rigidity Spectrometer), will extend the systematic study of quenching and correlation effects to the most neutron-rich nuclei accessible, testing whether the asymmetry dependence of spectroscopic factors (Section 19.4.4) persists to extreme isospin.

RIKEN's SAMURAI spectrometer at the Radioactive Isotope Beam Factory in Wako, Japan, has been at the forefront of knockout reaction spectroscopy for exotic nuclei. Recent highlights include the first spectroscopy of ${}^{39}\text{Na}$ (at the $N = 28$ shell closure for $Z = 11$) and systematic knockout measurements across the calcium isotopic chain from ${}^{48}\text{Ca}$ to ${}^{54}\text{Ca}$, revealing the emergence of a new sub-shell closure at $N = 34$ (Steppenbeck et al., 2013, Nature 502:207-210). The combination of RIKEN's intensity frontier and FRIB's isotope reach means that the next decade will see the single-particle structure of hundreds of previously inaccessible nuclei mapped for the first time.

19.5.4 Shell Evolution Far from Stability

Direct reactions at radioactive beam facilities have revealed several dramatic changes in nuclear structure far from stability:

1. New magic numbers. The $(d,p)$ and knockout experiments on oxygen isotopes showed that $N = 16$ becomes a new magic number in ${}^{24}\text{O}$, caused by the tensor force (Chapter 3) reducing the $\nu d_{3/2}$ - $\nu s_{1/2}$ gap when the proton $\pi d_{5/2}$ orbit is emptied. This was confirmed by the observation that ${}^{24}\text{O}$ is the last bound oxygen isotope (the neutron drip line).

2. Disappearing magic numbers. The $N = 20$ magic number disappears in the "island of inversion" around ${}^{32}\text{Mg}$ ($Z = 12$). Knockout reactions at MSU showed that the ground state of ${}^{32}\text{Mg}$ has a large component of neutron excitations across the $N = 20$ gap, giving it a deformed ground state rather than the spherical shape expected for a magic number.

3. The tensor force drives shell evolution. The systematic trends observed in single-particle energies from $(d,p)$ reactions across long isotopic chains are now understood as a consequence of the tensor component of the nuclear force (Otsuka et al., 2005, 2020). When a specific proton orbit is filled or emptied, the tensor interaction shifts the neutron single-particle energies in a characteristic pattern that can close old shell gaps and open new ones. Direct reactions provide the data that tests these predictions.

🔗 Connection to Chapter 6: The shell model predicts single-particle energies from a mean-field potential. Direct reactions measure those energies. The evolution of shell structure far from stability — new magic numbers appearing, old ones vanishing — is one of the most active and consequential frontiers in nuclear physics, with direct implications for $r$-process nucleosynthesis (Chapter 23) and the structure of neutron-rich matter in neutron stars (Chapter 25).

19.6 Summary and Connections

Direct reactions are the experimental foundation of nuclear single-particle structure. The key ideas of this chapter are:

1. Direct reactions are fast and peripheral. They happen in $\sim 10^{-22}$ s, involve surface nucleons, and carry specific quantum-state information in their angular distributions.

2. Stripping reactions $(d,p)$ determine $l$-values. The angular distribution of the outgoing proton reveals the orbital angular momentum of the transferred neutron, directly identifying the shell-model orbit.

3. Pickup reactions $(p,d)$ probe hole states. By removing a nucleon, they reveal the occupied orbits below the Fermi surface.

4. Knockout reactions $(e,e'p)$, $(p,2p)$ measure momentum distributions and spectroscopic factors. The electron-induced knockout has the advantage of a well-understood probe; the proton-induced knockout can be applied to radioactive beams.

5. The DWBA provides the theoretical framework. It separates the reaction mechanism (distorted waves, optical potentials) from the nuclear structure (spectroscopic factors). The factorization $d\sigma/d\Omega = S \times \sigma^{sp}_{\text{DWBA}}(\theta)$ is the central result.

6. Spectroscopic factors are quenched. Measured values are 55–70% of the independent-particle prediction, reflecting correlations (short-range, long-range, tensor) beyond the mean field.

7. Radioactive beam facilities extend direct reactions to exotic nuclei. Inverse kinematics, combined with $(d,p)$ transfer and $(p,2p)$ knockout, has revealed shell evolution — new magic numbers, vanishing shell gaps, and the role of the tensor force — throughout the nuclear chart.

Looking ahead: The fission reaction (Chapter 20) represents a qualitatively different kind of nuclear process — the large-scale rearrangement of nuclear matter — but the optical model and reaction theory developed here carry over directly. In nuclear astrophysics (Chapter 22), the nuclear reaction rates that power stars depend on the same spectroscopic factors measured by the direct reactions of this chapter. And at facilities like FRIB, the direct reaction program is pushing toward the neutron drip line, where the shell structure of the most neutron-rich bound nuclei will be mapped for the first time.

Chapter 19 Notation Summary

Direct reactions bridge the abstract shell model and the laboratory. Every single-particle energy level in the chart of nuclides, every spectroscopic factor, every new magic number reported from a radioactive beam experiment — all of these rest on the physics of this chapter. The projectile grazes the nuclear surface, a nucleon is transferred, and the angular distribution of the outgoing particle whispers the quantum numbers of the orbit. Learning to hear that whisper is what makes a nuclear experimentalist.