Case Study 1 — The (d,p) Reaction: How We Map Nuclear Shell Structure

The Question

How do experimentalists determine, orbit by orbit, which quantum states neutrons occupy inside an atomic nucleus? How was the shell model — a theoretical prediction based on a mean-field potential — actually verified by experiment?

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

${}^{208}\text{Pb}$ is the heaviest stable doubly magic nucleus, with $Z = 82$ protons filling all proton orbits through the $Z = 82$ closure and $N = 126$ neutrons filling all neutron orbits through the $N = 126$ closure. Its ground state is $J^\pi = 0^+$: spherical, with zero angular momentum.

When a deuteron transfers its neutron to ${}^{208}\text{Pb}$, the neutron must go into the first available orbits above the $N = 126$ shell closure. The shell model predicts these orbits (in order of increasing energy) to be $2g_{9/2}$, $1i_{11/2}$, $1j_{15/2}$, $3d_{5/2}$, $4s_{1/2}$, $2g_{7/2}$, $3d_{3/2}$ — the orbits that fill the $N = 126$–$184$ major shell.

The beauty of this system is its simplicity: the target is a closed shell, so the final-state wavefunction of ${}^{209}\text{Pb}$ is, to excellent approximation, a single neutron in a well-defined orbit coupled to the ${}^{208}\text{Pb}$ core. The spectroscopic factors should be close to unity, and the $l$-assignments unambiguous.

The Experiment

The definitive (d,p) measurements on ${}^{208}\text{Pb}$ were performed by multiple groups over three decades. The most complete data set comes from deuteron beams at $E_d \approx 17$–$22$ MeV at facilities including Aldermaston (UK), Saclay (France), and Oak Ridge (USA). The experimental setup is conceptually simple:

1. Beam: A deuteron beam from a tandem Van de Graaff or cyclotron accelerator, with energy $E_d \approx 20$ MeV and intensity $\sim 100$ nA.

2. Target: A thin ${}^{208}\text{Pb}$ foil ($\sim 0.5$–$2$ mg/cm$^2$), either enriched ${}^{208}\text{Pb}$ or natural lead (52% ${}^{208}\text{Pb}$). The target must be thin enough that the outgoing protons do not lose significant energy, but thick enough to give measurable counting rates.

3. Detection: A magnetic spectrograph or a set of silicon surface barrier detectors measures the energy and angle of the outgoing protons. The energy resolution ($\sim 30$–$80$ keV) must be sufficient to separate the low-lying states of ${}^{209}\text{Pb}$.

4. Data: For each final state, the differential cross section $d\sigma/d\Omega$ is measured at angles from $\sim 5°$ to $\sim 90°$ in steps of $2.5°$–$5°$.

The Angular Distributions

The measured angular distributions tell the story. Consider three representative states:

The ground state ($J^\pi = 9/2^+$, $l = 4$, $2g_{9/2}$): The angular distribution rises to a peak at $\theta_{\text{CM}} \approx 20°$, then falls to a deep minimum around $30°$, rises to a secondary maximum near $40°$, falls again, and continues oscillating with decreasing amplitude. The cross section at the first maximum is about 5 mb/sr. The pattern is characteristic of $l = 4$: the distribution vanishes at $0°$ (as it must for $l \neq 0$), and the positions of the minima match the DWBA prediction for $l = 4$.

The $4s_{1/2}$ state ($J^\pi = 1/2^+$, $l = 0$, $E_x = 2.032$ MeV): The angular distribution peaks sharply at $0°$ and falls monotonically with increasing angle, with no minimum in the forward hemisphere. This is the unmistakable signature of $l = 0$ — the only value of $l$ that produces a maximum at $\theta = 0°$. The cross section at $0°$ is about 8 mb/sr.

The $3d_{5/2}$ state ($J^\pi = 5/2^+$, $l = 2$, $E_x = 1.567$ MeV): The angular distribution shows a small but finite cross section at $0°$ that rises to a peak at $\theta_{\text{CM}} \approx 12°$, falls to a minimum around $25°$, rises to a second maximum near $35°$, and then declines. Two clear minima in the forward hemisphere confirm $l = 2$.

The DWBA Analysis

For each angular distribution, the experimentalist performs a DWBA calculation:

1. Optical potentials: The entrance-channel ($d + {}^{208}\text{Pb}$) optical potential is determined by fitting elastic scattering of deuterons from ${}^{208}\text{Pb}$ at the same energy. A typical parameter set: $V_0 \approx 100$ MeV, $r_0 = 1.15$ fm, $a = 0.78$ fm, $W_D \approx 14$ MeV. The exit-channel ($p + {}^{209}\text{Pb}$) optical potential comes from proton elastic scattering on nearby nuclei at the appropriate exit energy.

2. Bound-state wavefunction: A Woods-Saxon potential ($r_0 = 1.25$ fm, $a = 0.65$ fm) with depth adjusted to reproduce the neutron separation energy for each state. For the ground state, $B_n = 3.937$ MeV.

3. DWBA calculation: Using a code like DWUCK or FRESCO, the single-particle DWBA cross section $\sigma^{sp}_{\text{DWBA}}(\theta)$ is computed for each assumed $l$-value.

4. Comparison: The calculated angular distribution is compared to the data. The correct $l$-value is identified by the shape of the distribution. The spectroscopic factor is extracted from the magnitude: $S = (d\sigma/d\Omega)_{\text{exp}} / (d\sigma/d\Omega)_{\text{DWBA}}^{sp}$ at the first maximum (or from a global fit).

The results for the low-lying states of ${}^{209}\text{Pb}$:

The spectroscopic factors are close to — but systematically less than — unity, reflecting the correlations discussed in Section 19.4.

What We Learn

This single experiment provides:

1. The single-particle spectrum above $N = 126$. The energies of the first seven orbits in the $N = 126$–$184$ shell are measured to $\sim 1$ keV precision.

2. Confirmation of the shell model. The ordering and quantum numbers of the orbits match the shell-model prediction (Chapter 6) — including the crucial role of the spin-orbit interaction in placing the $2g_{9/2}$ ($l = 4$, $j = l + 1/2$) below the $1i_{11/2}$ ($l = 6$, $j = l + 1/2$).

3. The magnitude of correlations. The spectroscopic factors, while close to unity for this nearly ideal closed-shell-plus-one system, show a systematic reduction that increases for higher-lying orbits (higher $l$, higher excitation energy). This hints at the fragmentation of single-particle strength by long-range and short-range correlations.

4. A benchmark for nuclear theory. The ${}^{209}$Pb single-particle energies serve as a calibration standard for nuclear structure calculations. Any theoretical framework — shell model, density functional theory, coupled-cluster — must reproduce these energies.

Combined with the pickup reaction ${}^{208}\text{Pb}(p,d){}^{207}\text{Pb}$ (which maps the neutron hole states below $N = 126$), the (d,p) data provides the complete single-particle landscape around the $N = 126$ Fermi surface. This is the experimental evidence on which the shell model stands.

Discussion Questions

1. Why is ${}^{208}\text{Pb}$ a better target for establishing the single-particle spectrum than, say, ${}^{120}\text{Sn}$ ($Z = 50$, $N = 70$)?

2. The spectroscopic factors in the table above decrease for higher-$l$ orbits. Propose a physical explanation for this trend. (Consider the radial extent of the wavefunction and its overlap with the nuclear interior, where correlations are strongest.)

3. If you had only the (d,p) data (particle states above $N = 126$) and not the (p,d) data (hole states below $N = 126$), what important information about the shell model would you be missing?

4. The experiment uses $E_d \approx 20$ MeV deuterons. What would change if you used 8 MeV deuterons instead? (Think about the Coulomb barrier for $d + {}^{208}\text{Pb}$ and the matching conditions for the transferred neutron.)