--- title: Direct Reactions and the Optical Model module: Nuclear Reactions moduleNumber: 8 lessonNumber: 3 order: 803 summary: > A complex optical potential replaces the many-body target by a single particle moving in an average field whose imaginary part removes flux into non-elastic channels, reproducing the diffraction pattern of elastic scattering. Direct reactions bypass the compound nucleus, transferring a nucleon in one step: stripping and pickup deposit or remove a single nucleon, the angle of the first peak in the distorted-wave angular distribution fixes the transferred orbital angular momentum, and its magnitude gives the spectroscopic factor. topics: [Nuclear Reactions] sources: - book: Krane ref: "Ch. 11 — Nuclear Reactions; §11.5 The Optical Model, §11.11 Direct Reactions" - book: Wong ref: "Ch. 8 — Nuclear Reactions; §8-2 Optical Model, §8-3 Direct Reactions" draft: false --- 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.[^krane-opt] ## 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, $$ U(r) = -V\,f(r) - iW\,g(r) + V_{\text{so}}(r)\,\vec L\cdot\vec S + V_C(r). $$ The real part $-V f(r)$ is an attractive well of the Woods-Saxon shape $$ f(r) = \frac{1}{1 + \exp[(r - R)/a]}, \qquad R = r_0 A^{1/3}, $$ with depth $V \approx 40$–$50\,\mathrm{MeV}$ and diffuseness $a \approx 0.65\,\mathrm{fm}$. It refracts the incident wave. The **imaginary part** $-iW g(r)$ 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. $$ % caption: The optical potential has an attractive real well that refracts the wave and an % absorptive imaginary part, surface-peaked at low energy, that removes flux into non-elastic % channels. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {radius}; \draw[->, black] (0,-2.4) -- (0,1.6) node[above, black!70] {potential}; % real part: Woods-Saxon well (negative), flat bottom then rise to zero \draw[acc, very thick] (0.2,-2.0) -- (2.6,-2.0) .. controls (3.4,-2.0) and (3.7,-0.2) .. (4.4,-0.05) .. controls (5.4,0.0) and (6.6,0.0) .. (7.6,0.0); \node[acc, anchor=north west, font=\scriptsize] at (0.4,-1.6) {real well}; % imaginary part: surface-peaked bump (drawn positive for visibility) \draw[black, very thick, densely dashed] (0.2,0.05) .. controls (2.2,0.05) and (3.0,0.1) .. (3.6,0.7) .. controls (3.9,1.05) and (4.1,1.25) .. (4.35,1.25) .. controls (4.6,1.25) and (4.9,0.6) .. (5.5,0.25) .. controls (6.2,0.1) and (7.0,0.05) .. (7.6,0.05); \node[black, anchor=south west, font=\scriptsize] at (4.4,1.05) {absorptive part}; \draw[black, dashed] (3.1,-2.4) -- (3.1,1.3); \node[anchor=north, black, font=\scriptsize] at (3.1,-2.4) {surface}; \end{tikzpicture} $$ 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 $$ \Delta\theta \approx \frac{\pi}{kR}, $$ 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 $W$. $$ % caption: Elastic scattering from a strongly absorbing nucleus gives a forward-peaked % diffraction pattern whose oscillations, spaced by roughly pi over kR, measure the nuclear % radius; partial transparency fills in the minima. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {angle}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {yield (log)}; % diffraction: forward peak then decaying oscillations \draw[acc, very thick] (0.2,4.0) .. controls (0.7,3.2) and (1.0,2.1) .. (1.4,2.1) .. controls (1.7,2.1) and (1.9,3.0) .. (2.2,3.0) .. controls (2.5,3.0) and (2.7,1.5) .. (3.1,1.5) .. controls (3.4,1.5) and (3.6,2.3) .. (3.9,2.3) .. controls (4.2,2.3) and (4.4,1.0) .. (4.8,1.0) .. controls (5.1,1.0) and (5.3,1.7) .. (5.6,1.7) .. controls (5.9,1.7) and (6.1,0.6) .. (6.5,0.6) .. controls (6.8,0.6) and (7.0,1.1) .. (7.4,1.05); \node[acc, anchor=north east, font=\scriptsize] at (1.4,3.9) {forward peak}; \node[black, anchor=north, font=\scriptsize] at (4.8,0.85) {interference minima}; \end{tikzpicture} $$ ## 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. $$ % caption: In a (d,p) stripping reaction the deuteron's neutron is deposited into a % single-particle orbital of the target while the proton continues forward, populating a % shell-model state of the residual nucleus. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % target \draw[thick] (3.0,0) circle (0.85); \node[black] at (3.0,0) {target}; % incoming deuteron (proton + neutron) \fill[black] (-0.4,0.55) circle (2.2pt); \fill[acc] (-0.15,0.35) circle (2.2pt); \node[black, anchor=south, font=\scriptsize] at (-0.4,0.7) {$p$}; \node[acc, anchor=north, font=\scriptsize] at (-0.15,0.2) {$n$}; \draw[->, black] (0.1,0.45) -- (1.9,0.35); % neutron captured into orbital (dashed shell) \draw[black, densely dotted] (3.0,0) circle (0.55); \fill[acc] (3.0,0.55) circle (2.2pt); \node[acc, anchor=west, font=\scriptsize] at (3.15,0.6) {orbital}; % outgoing proton \draw[->, black] (4.0,0.3) -- (5.8,0.9); \fill[black] (5.9,0.95) circle (2.2pt); \node[black, anchor=west, font=\scriptsize] at (6.05,0.95) {$p$ out}; \end{tikzpicture} $$ ## 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 $$ \vec q = \vec k_i - \vec k_f, $$ 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 $$ \theta_1 \approx \frac{\ell}{k_i R} \quad(\text{small angles}), $$ 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. $$ % caption: The transfer angular distribution for each orbital angular momentum has its first % maximum at a characteristic angle that grows with the transferred angular momentum; reading % off that angle assigns the orbital. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.0,0) node[right, black!70] {angle}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {yield}; % ell = 0: peak at zero degrees, monotone falling then small bump \draw[acc, very thick] (0.2,3.9) .. controls (0.8,3.0) and (1.4,1.6) .. (2.2,1.0) .. controls (2.9,0.6) and (3.6,0.5) .. (4.2,0.7) .. controls (4.7,0.85) and (5.2,0.6) .. (6.0,0.4) .. controls (6.7,0.28) and (7.2,0.22) .. (7.6,0.2); \node[acc, anchor=south west, font=\scriptsize] at (0.4,3.6) {lower orbital}; % ell = 2: peak shifted out to ~2 \draw[black, very thick, densely dashed] (0.2,0.6) .. controls (0.9,0.8) and (1.5,2.4) .. (2.1,2.4) .. controls (2.6,2.4) and (3.2,1.0) .. (3.9,0.7) .. controls (4.5,0.5) and (5.0,0.9) .. (5.6,0.9) .. controls (6.2,0.9) and (6.9,0.4) .. (7.6,0.3); \draw[black, dashed] (2.1,0) -- (2.1,2.4); \node[black, anchor=south, font=\scriptsize] at (2.1,2.45) {higher orbital}; \end{tikzpicture} $$ ## 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, $$ T_{fi} \propto \int \chi_f^{(-)\ast}(\vec k_f, \vec r)\,\phi_{n\ell j}(\vec r)\, \chi_i^{(+)}(\vec k_i, \vec r)\,\d^3 r, $$ where $\chi_i^{(+)}$ and $\chi_f^{(-)}$ are the distorted incoming and outgoing waves and $\phi_{n\ell j}$ 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** $$ \left(\frac{\d\sigma}{\d\Omega}\right)_{\text{exp}} = S\,\left(\frac{\d\sigma}{\d\Omega}\right)_{\text{DWBA}}. $$ 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. > **Result.** A single-nucleon transfer reaction determines three quantities of a final state: > its energy gives the single-particle binding energy, the angle of the first peak in the DWBA > angular distribution gives the transferred orbital angular momentum $\ell$ (hence the parity > and the allowed $j$), and the magnitude of the cross section gives the spectroscopic factor > $S$, the occupancy of the orbital. 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](/nuclear-physics/nuclear-reactions/compound-nucleus-resonances) average of the compound part, so a single average potential underlies both descriptions. [^krane-opt]: **Krane**, _Introductory Nuclear Physics_, §11.5 (The Optical Model) and §11.11 (Direct Reactions), with **Wong**, _Introductory Nuclear Physics_, §8-2–8-3 — the complex Woods-Saxon optical potential and its absorptive imaginary part, elastic diffraction, single-nucleon stripping and pickup, the momentum-matching rule $qR \approx \ell$ for the transferred angular momentum, and the DWBA spectroscopic factor. Optical-model parameters and transfer data are compiled by the NNDC, [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).