Case Study 2 — Direct Reactions with Radioactive Beams: Nuclear Structure in Inverse Kinematics

The Question

How do we measure the single-particle structure of nuclei that exist for only fractions of a second? Nuclei far from the valley of stability — the neutron-rich isotopes that are forged in supernovae and neutron star mergers — cannot be made into targets. How has the experimental technique of direct reactions been reinvented for the radioactive beam era?

The Landmark Experiment: ${}^{132}\text{Sn}(d,p){}^{133}\text{Sn}$

Why ${}^{132}\text{Sn}$?

${}^{132}\text{Sn}$ ($Z = 50$, $N = 82$) is a doubly magic nucleus: the neutron-rich counterpart of ${}^{208}\text{Pb}$. It lies far from the valley of stability — it is 18 neutrons beyond the heaviest stable tin isotope (${}^{124}\text{Sn}$) — and decays by $\beta^-$ emission with a half-life of 39.7 s. It cannot be made into a target foil.

${}^{132}\text{Sn}$ is scientifically crucial because:

1. It anchors the $N = 82$ shell closure far from stability. Is the $N = 82$ gap the same in ${}^{132}\text{Sn}$ as in the stable nucleus ${}^{138}\text{Ba}$ ($N = 82$, $Z = 56$)? If the gap changes, the $r$-process nucleosynthesis path shifts, altering predictions for the abundances of heavy elements in the universe.

2. The single-particle spectrum above $N = 82$ determines the structure of nuclei in the $A \approx 130$ region, which includes the $r$-process abundance peak near $A = 130$. These are the nuclei that freeze out of nuclear statistical equilibrium as the neutron flux drops in a neutron star merger.

3. It is the heaviest doubly magic nucleus accessible to direct reactions with radioactive beams using current technology.

The Experiment (Jones et al., 2010)

The experiment was performed at the Holifield Radioactive Ion Beam Facility (HRIBF) at Oak Ridge National Laboratory (ORNL) in Tennessee.

Beam production: ${}^{132}\text{Sn}$ was produced by proton-induced fission of uranium carbide targets in the ISOL (Isotope Separator On-Line) facility. The fission fragments were ionized, mass-separated, and accelerated to $E/A = 4.8$ MeV/nucleon using the ORNL tandem accelerator. The beam intensity was approximately $2 \times 10^4$ particles per second — roughly eight orders of magnitude less than a typical stable beam, but sufficient for the measurement.

Target: A deuterated polyethylene target (CD$_2$), approximately 100 $\mu$g/cm$^2$ thick. The deuterons in the target serve as the "projectile" in the reaction — even though, in the lab frame, they are at rest. A pure carbon target was also measured to subtract background from reactions on carbon.

Detection: The ORRUBA (Oak Ridge Rutgers University Barrel Array) — an array of silicon strip detectors arranged in a barrel geometry around the target — detected the outgoing protons. Because the center-of-mass frame is moving forward rapidly (the heavy ${}^{132}\text{Sn}$ beam carries most of the momentum), the protons emerge at backward lab angles. The ORRUBA was positioned to cover lab angles from approximately $85°$ to $170°$, corresponding to center-of-mass angles from $10°$ to roughly $50°$.

The heavy reaction residue (${}^{133}\text{Sn}$) was identified downstream by the Recoil Mass Spectrometer, confirming that the $(d,p)$ reaction had indeed occurred.

Results

The proton energy spectrum at backward lab angles showed clear peaks corresponding to different states of ${}^{133}\text{Sn}$:

The angular distributions, though measured over a limited angular range (a fundamental limitation of inverse kinematics), showed the characteristic $l$-dependent patterns. The $l = 3$ ground-state distribution peaked away from $0°$ with three oscillations visible; the $l = 1$ states peaked closer to $0°$ with a single minimum.

Comparison with ${}^{208}\text{Pb}$ Region

The most important result is the comparison of single-particle spacings between ${}^{133}\text{Sn}$ (above $N = 82$ in $Z = 50$) and ${}^{209}\text{Pb}$ (above $N = 126$ in $Z = 82$):

The spin-orbit splitting between the $2f_{7/2}$ and $2f_{5/2}$ orbits in ${}^{133}\text{Sn}$ is $2.005$ MeV. Shell-model predictions with modern effective interactions (including the tensor force) had predicted values in the range 1.7–2.2 MeV. The measurement confirmed that the spin-orbit splitting evolves relatively slowly with asymmetry in this region — important for $r$-process calculations.

Technical Challenges of Inverse Kinematics

This experiment illustrates the challenges that define radioactive beam physics:

1. Low beam intensity. At $2 \times 10^4$ pps, the experiment required roughly one week of continuous beam time to accumulate a few hundred counts per angular distribution point. Compare this to a stable-beam experiment that might accumulate thousands of counts per point in an hour.

2. Limited angular range. In inverse kinematics, the CM-to-lab transformation compresses the forward CM angles into backward lab angles and vice versa. The detector array covers only a portion of the CM angular range, and the angular resolution is degraded by the kinematic compression. Full angular distributions, as measured in normal kinematics, are often impossible.

3. Background subtraction. The CD$_2$ target contains both deuterium and carbon. Reactions on ${}^{12}\text{C}$ produce protons at similar energies and must be subtracted using the pure carbon target measurement.

4. Beam purity. The ISOL beam contains not only ${}^{132}\text{Sn}$ but also isobaric contaminants (nuclei with the same mass $A = 132$ but different $Z$). Beam purification techniques — resonant laser ionization, isobar separators — are essential.

5. Target effects. The thin target ($100$ $\mu$g/cm$^2$) causes energy loss and straggling of both the beam and the protons, degrading energy resolution. Thicker targets increase the count rate but worsen the resolution.

Despite these challenges, the experiment succeeded in extracting the single-particle spectrum of ${}^{133}\text{Sn}$ — data that had been sought for decades and that no other technique could provide.

Impact on Nuclear Astrophysics

The ${}^{132}\text{Sn}(d,p)$ result has direct implications for the $r$-process (Chapter 23). The $r$-process path passes through or near ${}^{132}\text{Sn}$ at the $N = 82$ "waiting point," where the neutron capture rate temporarily stalls because of the large shell gap. The rate of neutron capture past $N = 82$ — and therefore the production of elements heavier than tellurium and xenon — depends on the energies of the single-particle orbits above $N = 82$ in this region.

Before the ORNL experiment, $r$-process models used single-particle energies extrapolated from stable nuclei or predicted by various mass models, with uncertainties of several hundred keV. The measured values reduced these uncertainties to tens of keV for the key orbits, sharpening the $r$-process abundance predictions in the $A \approx 130$ peak.

The Future: FRIB and Beyond

The ${}^{132}\text{Sn}(d,p)$ experiment was a proof of principle. The Facility for Rare Isotope Beams (FRIB), which began operations at Michigan State University in 2022, produces radioactive beams at intensities 100–1000 times higher than HRIBF for many neutron-rich isotopes. This enables:

• Transfer reactions on nuclei even further from stability (e.g., ${}^{130}\text{Cd}$, ${}^{134}\text{Sn}$)
• Higher statistics, allowing measurement of weaker states (smaller spectroscopic factors)
• Knockout reactions at intermediate energies for systematic spectroscopy across long isotopic chains

The combination of direct reactions and radioactive beams is mapping the single-particle landscape across the nuclear chart — testing the shell model in regions where its predictions are most uncertain, and providing the nuclear physics input that astrophysical models need to explain the origin of the heavy elements.

Discussion Questions

1. Why was it necessary to perform this experiment in inverse kinematics? Why not simply make a ${}^{132}\text{Sn}$ target?

2. The beam intensity of $2 \times 10^4$ pps is roughly $10^8$ times less than a stable beam. How does this affect the experimental strategy (target thickness, measurement time, statistical precision)?

3. The angular distributions measured in inverse kinematics cover a limited range. How does this affect the reliability of the $l$-assignments compared to normal-kinematics measurements on stable nuclei?

4. Why is the $N = 82$ shell gap in ${}^{132}\text{Sn}$ particularly important for $r$-process nucleosynthesis? What would happen to the $r$-process abundance pattern if the $N = 82$ gap were smaller in ${}^{132}\text{Sn}$ than in stable nuclei?

5. Compare the experimental approach at an ISOL facility (like HRIBF, producing low-energy reaccelerated beams) with that at an in-flight fragmentation facility (like RIKEN or FRIB, producing fast beams at $E/A \sim 200$ MeV/nucleon). What reactions are possible at each, and what are the trade-offs?