Direct Reactions and the Optical Model

Not every reaction equilibrates through a compound nucleus. When the projectile interacts with only a few nucleons and leaves before its energy is shared, the reaction is direct: it proceeds on the transit timescale $\sim 10^{-22}\mathrm{s}$, its products are strongly forward-peaked, and its cross section varies smoothly with energy rather than resonating. The tool for both the elastic channel and these one-step transfers is the optical model, which replaces the intractable many-body target by an average complex potential.¹

The optical potential

Model the projectile as a single particle moving in a complex potential that stands in for its average interaction with all $A$ target nucleons,

The real part $-Vf(r)$ is an attractive well of the Woods-Saxon shape

with depth $V\approx 40$–$50\mathrm{MeV}$ and diffuseness $a\approx 0.65\mathrm{fm}$. It refracts the incident wave. The imaginary part $-iWg(r)$ is the essential addition: it removes probability flux from the elastic channel, representing the many ways the projectile can be absorbed into inelastic, transfer, or compound channels. At low energy the absorption is concentrated at the nuclear surface, so $g(r)$ is often taken as the derivative of a Woods-Saxon form, peaked near $r=R$. The spin-orbit term $V_{\text{so}}\vec{L}\cdot \vec{S}$ polarizes the scattered beam, and $V_C$ is the Coulomb potential of the charge distribution.

Solving the Schrödinger equation with this complex $U(r)$ yields a non-unitary scattering matrix: the elastic wave emerges with reduced amplitude, and the missing flux is the total reaction cross section. A single set of energy-dependent parameters $(V,W,r_0,a)$ fits the elastic scattering and reaction cross sections across a wide range of nuclei, which is why the model is called optical — the nucleus behaves like a refracting, partially absorbing sphere, a cloudy crystal ball.

Elastic diffraction

Because the absorbing nucleus has a sharp surface of radius $R$ comparable to the projectile wavelength, elastic scattering shows a diffraction pattern like light past an opaque disk. The angular distribution oscillates, with maxima spaced by

where $k$ is the incident wavenumber. Strong absorption (a black nucleus) produces Fraunhofer diffraction with deep, regularly spaced minima; partial transparency softens them into the Fresnel pattern of a grey nucleus. The spacing of the minima measures the nuclear radius, and the depth of the minima measures the strength of the absorption $W$.

Direct reactions and transfer

A direct reaction transfers one or a few nucleons in a single step, without forming an equilibrated intermediate. The clearest cases are single-nucleon transfers:

• Stripping, such as $(d,p)$: an incident deuteron passes the target, its neutron is captured into a single-particle orbital, and the proton continues nearly undeflected. The target gains a neutron.
• Pickup, such as $(p,d)$: an incident proton captures a neutron out of an occupied target orbital and leaves as a deuteron. The target loses a neutron.

Both are the inverse of each other, and both populate specific final states: stripping reaches states describable as the target ground state plus a nucleon in a definite orbital, while pickup reaches states describable as the target minus a nucleon from a definite orbital. The final-state energy fixes the single-particle binding energy of that orbital, so a transfer spectrum maps the shell-model levels directly.

Angular momentum from the angular distribution

The power of transfer reactions comes from the angular distribution of the outgoing particle. In the plane-wave picture the transferred nucleon carries momentum

the difference of the incident and outgoing wavevectors, and it must be deposited into an orbital of orbital angular momentum $\ell$ at the nuclear radius $R$. Matching the transferred momentum to the orbital angular momentum, $qR\approx \ell$, ties the momentum transfer to $\ell$. Since $q$ grows with scattering angle, the differential cross section peaks at an angle that increases with $\ell$: the first maximum sits near

so measuring the angle of the first peak fixes the transferred orbital angular momentum, and with it the parity change of the final state. The distinct diffraction-like pattern for each $\ell$ is the fingerprint that assigns spins and parities to shell-model states.

Distorted waves and spectroscopic factors

Plane waves overstate the pattern; the projectile and ejectile are distorted by the optical potentials of the entrance and exit channels. The distorted-wave Born approximation (DWBA) replaces the plane waves by optical-model scattering wavefunctions and computes the transfer amplitude as an overlap integral,

where ${\chi }_i^{(+)}$ and ${\chi }_f^{(-)}$ are the distorted incoming and outgoing waves and ${\varphi }_{n\ell j}$ is the bound-state wavefunction of the transferred nucleon. The DWBA cross section reproduces the shape of the angular distribution, and its overall magnitude, compared with experiment, defines the spectroscopic factor

The spectroscopic factor $S$ measures how completely the final state is described by the target plus a nucleon in the orbital $n\ell j$: it is the occupancy (for pickup) or vacancy (for stripping) of that orbital. A pure single-particle state has $S$ near unity; configuration mixing spreads the single-particle strength over several states, and the sum of $S$ over them obeys a sum rule set by the shell occupancy. Transfer reactions analyzed this way are the primary experimental source of single-particle energies and occupancies in the shell model.

Direct and compound mechanisms are limiting cases of one reaction. At a given energy both can contribute: the smooth, forward-peaked direct part rides on the symmetric, isotropic compound part, and the two are separated by their angular distributions and their energy dependence. The optical model supplies the distorted waves for the direct calculation and the transmission coefficients for the Hauser-Feshbach average of the compound part, so a single average potential underlies both descriptions.