Inelastic Scattering and Multichannel Theory

Slide a control rod into a reactor and the neutron population collapses — nothing bounced those neutrons away; cadmium swallowed them. Yet a stubborn wave-mechanical theorem says swallowing cannot be the whole story: whatever absorbs a quantum beam is forced, by unitarity alone, to scatter part of it elastically. An absorber with no echo is impossible. This article opens the doors elastic theory kept shut. A real target — atom, nucleus, molecule — has internal states, so a collision can excite it, ionize it, or swallow the projectile. Each distinct final configuration is a channel, and every object of single-channel scattering is promoted to a matrix in channel space. We derive the bookkeeping that tames this: the entrance-channel element shrinks to $S_\ell=\eta_\ell e^{2i\delta_\ell}$ with inelasticity $\eta_\ell\le 1$ , and the flux ledger splits into elastic $\propto|1-S_\ell|^2$ and reaction $\propto(1-\eta_\ell^2)$ cross sections. From that ledger we prove the no-free-absorption theorem ( $\sigma_{\mathrm{inel}}>0$ forces $\sigma_{\mathrm{el}}>0$ ), work out the black-disk limit where shadow and absorption each lock onto $\pi R^2$ , and solve a two-channel model exactly — unitarity holding to machine precision, and a Wigner cusp snapping into the elastic cross section the instant a new channel opens. Run on real numbers, the machinery holds a surprise: cadmium's monstrous 20,600-barn appetite spends less than a tenth of a percent of the s-wave unitarity budget — and its unavoidable elastic echo is a few barns, as measured.

Key takeaways

A channel = one asymptotic internal configuration, with threshold $\epsilon_n$ and wavenumber $k_n=\sqrt{2\mu(E-\epsilon_n)}/\hbar$ — open if $E>\epsilon_n$ , closed (decaying) otherwise.
The full channel-space S-matrix stays unitary, but the entrance element alone obeys $|S_\ell|=\eta_\ell\le1$ : the missing modulus is flux that left through other doors.
Flux bookkeeping gives $\sigma_{\mathrm{el}}=\frac{\pi}{k^2}\sum(2\ell+1)|1-S_\ell|^2$ and $\sigma_{\mathrm{inel}}=\frac{\pi}{k^2}\sum(2\ell+1)(1-\eta_\ell^2)$ — derived here, verified symbolically.
No free absorption: $|1-S_\ell|^2\ge(1-\eta_\ell)^2$ — absorbing waves must scatter. Black-disk extreme: $\sigma_{\mathrm{el}}=\sigma_{\mathrm{inel}}=\pi R^2$ , total $2\pi R^2$ .
A two-channel square well solved exactly: $S^\dagger S=\mathbb 1$ to $10^{-12}$ , and a threshold cusp in $\sigma_{\mathrm{el}}$ where channel 2 opens; a closed-channel bound state near threshold = the Feshbach mechanism.
Key idea: replace the scalar S-matrix by a unitary matrix in channel space; the entrance-channel element $S_\ell=\eta_\ell e^{2i\delta_\ell}$ with $\eta_\ell\le1$ splits the flux into elastic and reaction parts, and the forward elastic amplitude still pays for all of it.

The core insight

Replace the scalar S-matrix by a unitary matrix in channel space; the entrance-channel diagonal element $S_\ell=\eta_\ell e^{2i\delta_\ell}$ with $\eta_\ell\le1$ splits flux into elastic and reaction parts, and the optical theorem ties the total to the forward elastic amplitude.

Where this lives today

Every ultracold-atom experiment that "tunes interactions" runs this article's physics in a magnetic field: a closed hyperfine channel's bound state is Zeeman-shifted across threshold and the scattering sweeps with it — unitary Fermi gases, Efimov physics, molecular condensates are all Feshbach engineering. In nuclear technology, the $1/v$ law and the optical model remain the working language of reactor design; evaluated nuclear data files tabulate, in effect, $\eta_\ell(E)$ and $\delta_\ell(E)$ for every isotope that matters. And at colliders, the elastic-to-total ratio climbing from 0.25 toward the black-disk 0.5 is a live diagnostic of the proton's growing opacity — Example 2's grey disk, remeasured at every new beam energy.

The physical picture

Picture the collision as a hallway with many exit doors: elastic, target excited, target ionized, projectile captured. Every door is a channel, labelled by the internal state left behind, and energy conservation prices each ticket — a channel costing $\epsilon_n$ opens only when the collision energy can pay it. Below that price the door is closed: its wavefunction component decays exponentially and carries no flux out. But a closed door is not an absent door — the system can lean on it at short range, and if a bound state hides just behind it, the leaning becomes a resonance.

Figure

Channels: one entrance, many priced exits

Unitarity is door conservation: everything that goes in comes out somewhere. Watch only the elastic door and you see less return — that deficit is the inelasticity $\eta_\ell<1$ . Geometrically this lives on the Argand circle established in The S-Matrix and Unitarity in Scattering: an elastic wave can only rotate its outgoing phase, pinning $S_\ell$ to the unit circle; open reaction channels pull the point inside. This article owns the quantitative ledger of that retreat — and its sharpest consequence, in the punchline below: an absorber can never be echo-free, and a perfect absorber echoes exactly as much as it eats.

Figure

Black disk: absorption and shadow each lock onto πR² — and the Argand point falls inside

An imaginary potential is the cartoon of all this: with $U=V-iW$ , $W>0$ , the continuity equation grows a sink — probability drains at rate $2W|\psi|^2/\hbar$ , not destroyed but relocated into channels we chose not to track. Conservation of Normalization introduced this device for decaying states ( $V-i\Gamma/2$ , lifetime $\tau=\hbar/\Gamma$ ); here it becomes nuclear physics' optical model, a complex well summarizing hundreds of channels in two fitted functions.

The channel ledger

Upgrade the asymptotic wavefunction of Asymptotic Form of the Scattering Wave Function to carry internal labels: a beam enters in state $|a\rangle$ , and far away the stationary state describes every allowed outcome at once:

\psi \xrightarrow{r\to\infty}\; e^{ik_a z}\,|a\rangle \;+\; \sum_{b\ \mathrm{open}}\, f_{ba}(\theta)\,\frac{e^{ik_b r}}{r}\,|b\rangle \qquad (1)

where: $|a\rangle,|b\rangle$ — entrance and exit internal states; $f_{ba}(\theta)$ — the transition amplitude; $k_b=\sqrt{2\mu(E-\epsilon_b)}/\hbar$ — the exit wavenumber fixed by energy conservation. Channels with $\epsilon_b>E$ have imaginary $k_b$ : they decay exponentially and deliver nothing to the detector — closed doors. Equation (1) says a multichannel experiment is several scattering experiments in superposition, all fed by one entrance wave. Because the exit channel moves at its own speed, a transition cross section picks up a velocity ratio — the flux-ratio construction of Differential and Total Cross Section, applied per doorway ( $j_b=(\hbar k_b/\mu)|f_{ba}|^2/r^2$ against incident $\hbar k_a/\mu$ ):

\frac{\mathrm{d}\sigma_{a\to b}}{\mathrm{d}\Omega} = \frac{k_b}{k_a}\,\bigl|f_{ba}(\theta)\bigr|^2 \qquad (2)

where: the prefactor $k_b/k_a$ is unity for elastic scattering, below one for endothermic excitation (products crawl away slower than the beam arrived). Forgetting it is the classic multichannel bookkeeping error — it belongs to the counting, not the amplitude. For a central interaction each partial wave scatters independently, and all the new physics condenses into one complex number per wave. The elastic element of the channel-space S-matrix is written

S_\ell \;=\; \eta_\ell\, e^{2i\delta_\ell},\qquad 0\le \eta_\ell\le 1 \qquad (3)

where: $\delta_\ell$ — the real phase shift of Phase Shifts; $\eta_\ell$ — the inelasticity, the surviving modulus. The unit-circle geometry of The S-Matrix and Unitarity in Scattering — elastic scattering can only rotate the outgoing wave, never shrink it — holds while every reaction door is shut; the moment one opens, the partial-wave point falls inside the circle. The elastic amplitude keeps its form from Partial-Wave Expansion, with $e^{2i\delta_\ell}\to\eta_\ell e^{2i\delta_\ell}$ :

f_{\mathrm{el}}(\theta) = \frac{1}{2ik}\sum_{\ell=0}^{\infty}(2\ell+1)\,\bigl[\eta_\ell e^{2i\delta_\ell}-1\bigr]\,P_\ell(\cos\theta) \qquad (4)

where: $k\equiv k_a$ throughout; the $-1$ still subtracts the unscattered beam. Everything measurable now follows from how $|S_\ell|<1$ redistributes flux — the first derivation's job.

Derivation APartition of the cross section into elastic and reaction parts

We derive the central ledger of multichannel scattering: how one complex number $S_\ell=\eta_\ell e^{2i\delta_\ell}$ per wave splits the total cross section into elastic and reaction pieces. The strategy is radial flux bookkeeping — write the $\ell$ -th wave as incoming plus outgoing spherical waves, let $S_\ell$ rescale only the outgoing one, and count what fails to return. The corollary — absorption forces elastic scattering — is the article's sharpest theorem; watch it drop out of a two-line inequality.

Step 1. Far from the target, the $\ell$ -th partial wave is a superposition of incoming and outgoing spherical waves,

u_\ell(r) \;\propto\; e^{-i(kr-\ell\pi/2)} \;-\; S_\ell\, e^{+i(kr-\ell\pi/2)},

with the same incoming coefficient as the free plane wave — the beam is fixed by the experiment; the target can only edit what comes back out. This in/out split of the plane wave is Partial Wave Expansion of Plane Waves' ledger reading: the free beam is a perfectly balanced account of arrivals and departures in every channel, and scattering can only re-book the departures.

Step 2. The net probability flux removed from the elastic channel in wave $\ell$ is the incoming flux minus the surviving outgoing flux, proportional to $1-|S_\ell|^2=1-\eta_\ell^2$ . Dividing by the incident flux and attaching the standard partial-wave weight $\pi(2\ell+1)/k^2$ — the maximal cross section a single wave can offer — gives the reaction cross section:

\sigma_{\mathrm{inel}} \;=\; \frac{\pi}{k^2}\sum_{\ell}(2\ell+1)\,\bigl(1-\eta_\ell^2\bigr).

Why $1-\eta_\ell^2$ : fluxes go as squared moduli, and full unitarity makes the deficit $1-|S_\ell^{aa}|^2$ exactly $\sum_{b\ne a}|S_\ell^{ba}|^2$ — probability relocated through other doors, never lost.

Step 3. The elastic cross section counts what returns redirected. Integrating $|f_{\mathrm{el}}|^2$ over angles, Legendre orthogonality collapses the double sum:

\sigma_{\mathrm{el}} \;=\; \frac{\pi}{k^2}\sum_{\ell}(2\ell+1)\,\bigl|1-\eta_\ell e^{2i\delta_\ell}\bigr|^2 \;=\; \frac{\pi}{k^2}\sum_{\ell}(2\ell+1)\,\bigl(1-2\eta_\ell\cos 2\delta_\ell+\eta_\ell^2\bigr).

Step 4. Add the two ledger columns. The $\eta_\ell^2$ terms cancel between them:

\sigma_{\mathrm{tot}} \;=\; \sigma_{\mathrm{el}}+\sigma_{\mathrm{inel}} \;=\; \frac{2\pi}{k^2}\sum_{\ell}(2\ell+1)\,\bigl(1-\eta_\ell\cos 2\delta_\ell\bigr).

Step 5. The corollary — no free absorption. Expand the elastic summand about its floor:

\bigl|1-\eta_\ell e^{2i\delta_\ell}\bigr|^2 \;=\; (1-\eta_\ell)^2 \;+\; 2\eta_\ell\,(1-\cos 2\delta_\ell)\;\ge\;(1-\eta_\ell)^2.

If wave $\ell$ absorbs at all ( $\eta_\ell<1$ ), its elastic contribution is at least $\frac{\pi}{k^2}(2\ell+1)(1-\eta_\ell)^2>0$ — even at $\delta_\ell=0$ . You cannot remove flux from a wave without diffracting some of it. A stealth target that only absorbs is forbidden by unitarity. Verified: all four identities checked symbolically (sympy) in verify_aq-qm-inelastic-multichannel-scattering.py — the expansion of $|1-\eta e^{2i\delta}|^2$ , the per-wave sum $\mathrm{el}+\mathrm{inel}=2(1-\eta\cos2\delta)$ , the $\eta=1$ elastic limit $4\sin^2\delta$ , and the floor inequality scanned numerically over the full $(\eta,\delta)$ rectangle.

Consistency closes the loop: evaluating equation (4) forward ( $P_\ell(1)=1$ ) gives $\mathrm{Im}\,f_{\mathrm{el}}(0)=\frac{1}{2k}\sum(2\ell+1)(1-\eta_\ell\cos2\delta_\ell)$ , and $4\pi/k$ times this reproduces Step 4's $\sigma_{\mathrm{tot}}$ including the reaction part. That is the statement of the Optical Theorem — the forward elastic amplitude pays for all flux removed, absorbed or scattered. Its derivation belongs to that article; here the multichannel ledger is exhibited (symbolically) to honor it line by line.

Two channels, solved exactly

Coupled channels sound abstract until you solve one. Projecting the Schrödinger equation on the internal states gives coupled radial equations,

\Bigl[-\frac{\hbar^2}{2\mu}\Bigl(\frac{\mathrm{d}^2}{\mathrm{d}r^2}-\frac{\ell(\ell+1)}{r^2}\Bigr)+\epsilon_n-E\Bigr]u_n(r)+\sum_{m}V_{nm}(r)\,u_m(r)=0 \qquad (5)

where: $u_n$ — the radial function in channel $n$ ; $V_{nm}=\langle n|\hat V|m\rangle$ — the coupling matrix. Diagonal entries scatter within a channel; off-diagonal entries move probability between channels — the only way $\eta_\ell$ can drop below one. For square wells the system is exactly solvable:

Go deeper · Derivation B · 1 minA two-channel square well: exact S-matrix, unitarity, and the threshold cusp

Derivation BA two-channel square well: exact S-matrix, unitarity, and the threshold cusp

We build the smallest honest multichannel problem and extract its full $2\times2$ S-matrix to machine precision, with three payoffs: unitarity emerges numerically exact rather than assumed; $\eta_1$ switches on the instant channel 2 opens; and the elastic cross section develops a Wigner cusp — a genuine kink — at threshold. Units $\hbar=2\mu=1$ ; s-wave; thresholds $\epsilon_1=0,\ \epsilon_2=2$ ; inside $r<a=1$ a constant coupling matrix with $V_{11}=-4$ , $V_{22}=-1.5$ , $V_{12}=0.9$ ; free, uncoupled outside.

Step 1. Inside the well, (5) reads $\mathbf u''=W\mathbf u$ with constant $W=V+\mathrm{diag}(\epsilon)-E\mathbb 1$ . Diagonalize: in $W$ 's eigenbasis the channels decouple into free waves with interior wavenumbers $\tilde k_j=\sqrt{-\lambda_j}$ ; the regular solutions are $\sin(\tilde k_j r)$ rotated back by the eigenvector matrix $P$ . Why diagonalizing works: only because $V$ is constant in $r$ . Radial-dependent couplings mix the eigenbasis at every radius — the case close-coupling codes integrate numerically; the threshold physics is identical.

Step 2. Outside, each channel is free: $u_n = A_n e^{-ik_nr}-B_ne^{+ik_nr}$ , $k_n=\sqrt{E-\epsilon_n}$ (imaginary $k_2$ below threshold — the closed channel handles itself in the same algebra). Matching the interior log-derivative matrix $R=U(a)U'(a)^{-1}$ to these forms, with flux normalization $\sqrt{k_n}$ per channel, yields the exact S-matrix

S \;=\; \mathrm{diag}(\sqrt{k})\;\bigl(O-RO'\bigr)^{-1}\bigl(I-RI'\bigr)\;\mathrm{diag}(1/\sqrt{k}),

with $I,O$ the incoming/outgoing exponentials at $r=a$ .

Step 3. Run the numbers. At $E=3.5$ (both open): $\|S^\dagger S-\mathbb 1\|=8\times10^{-16}$ — unitarity to machine precision, assumed nowhere. The identity $1-\eta_1^2=|S_{12}|^2$ holds to $10^{-16}$ : the elastic deficit is the transition probability. At $E=1.2$ (channel 2 closed): $|S_{11}|=1$ exactly — a closed door restores the unit circle while still distorting $\delta_0$ .

Step 4. Scan the threshold. For $E<\epsilon_2$ , $\eta_1=1$ to $10^{-10}$ ; above it, $\eta_1<1$ immediately (deficit $2\times10^{-2}$ by $E=2.4$ ). The elastic cross section $(\pi/k_1^2)|1-S_{11}|^2$ has one-sided slopes $+8.9$ below and $-84.0$ above threshold — a cusp, not a smooth maximum. The new channel's flux grows as $k_2\propto\sqrt{E-\epsilon_2}$ (Wigner's law), stolen from the elastic channel with infinite initial slope. Verified: every claim is a numbered check in verify_aq-qm-inelastic-multichannel-scattering.py — unitarity deviation $8\times10^{-16}$ , sub-threshold $|S_{11}|-1$ at $10^{-16}$ , $\eta_1$ deficit scan, one-sided slopes $+8.89/-83.99$ , and the $1-\eta_1^2=|S_{12}|^2$ identity.

The same picture explains the Feshbach resonance. Let the closed channel hold a bound state at $E_{\mathrm{res}}$ just above the open threshold: a colliding pair borrows it virtually — enters, lingers, returns — and the open channel acquires a resonance at $E\approx E_{\mathrm{res}}$ with width set by $|V_{12}|^2$ . A magnetic field Zeeman-shifts the closed channel, so the resonance can be steered — the tuning knob of ultracold physics. The scattering-length consequences belong to Low-Energy Scattering and the Scattering Length and Bound States and Zero-Energy Resonances; the multichannel mechanism behind the knob lives here. The peak's lineshape is Resonances and the Breit-Wigner Formula's story.

The optical potential

When dozens of channels are open, tracking them all is hopeless and unnecessary. The optical model replaces the untracked channels by a complex potential,

U_{\mathrm{opt}}(r) = V(r) - i\,W(r),\qquad W(r)\ge 0 \qquad (6)

where: $V$ refracts and $W$ absorbs. In the continuity equation $U_{\mathrm{opt}}$ adds a sink $-(2W/\hbar)|\psi|^2$ : probability drains wherever $W>0$ (verified symbolically for constant $W$ : the norm decays as $e^{-2Wt/\hbar}$ ). This is the scattering sibling of the decaying-state device $V-i\Gamma/2$ of Conservation of Normalization. Fitted Woods–Saxon $V,W$ reproduce nucleon–nucleus elastic data across the periodic table; the fitted $W$ is a smooth summary of every open channel, and $\eta_\ell<1$ per wave reads straight back out. The sign matters: $W<0$ would create probability — impossible for a passive target.

Limiting cases

Case	Result	Meaning
All doors shut	$\eta_\ell=1$ , $S_\ell=e^{2i\delta_\ell}$	Single-channel theory recovered exactly; the Argand point returns to the unit circle.
Partial absorption	$0<\eta_\ell<1$	Both cross sections nonzero; elastic floor $(\pi/k^2)(2\ell+1)(1-\eta_\ell)^2$ survives even at $\delta_\ell=0$ .
Black disk, $kR\gg1$	$\sigma_{\mathrm{el}}=\sigma_{\mathrm{inel}}=\pi R^2$	Perfect absorber echoes exactly what it eats; $\sigma_{\mathrm{tot}}=2\pi R^2$ (derivation below).
Exothermic s-wave, $k\to0$	$\sigma_{\mathrm{inel}}\propto 1/k \propto 1/v$	Wigner threshold law: the $1/v$ absorption behind reactor control rods and neutron detection.
New channel opens	$1-\eta_1^2\propto\sqrt{E-\epsilon_2}$	Square-root onset ⇒ cusp in $\sigma_{\mathrm{el}}$ at every threshold (exhibited exactly above).

Go deeper · Derivation C · 1 minThe black disk: absorption πR², shadow πR², total 2πR²

Derivation CThe black disk: absorption πR², shadow πR², total 2πR²

We derive the extreme case of the no-free-absorption theorem: a totally absorbing sphere at $kR\gg1$ scatters elastically exactly as much as it absorbs. The route: feed the crudest $\eta_\ell$ profile into the partition theorem and do the finite sums honestly — the discrete sums are exact and expose the convergence rate too.

Step 1. Semiclassically, wave $\ell$ probes impact parameter $b\approx\ell/k$ — the dictionary $\ell+\tfrac12=kb$ of Partial Wave Expansion of Plane Waves. Total absorption for $b<R$ , nothing beyond:

\eta_\ell=0,\ \delta_\ell=0 \quad (\ell\le L\equiv\lfloor kR\rfloor), \qquad \eta_\ell=1,\ \delta_\ell=0 \quad (\ell>L).

Step 2. Reaction column of the ledger: $\sigma_{\mathrm{inel}}=\frac{\pi}{k^2}\sum_{0}^{L}(2\ell+1)\cdot 1 = \frac{\pi}{k^2}(L+1)^2 \to \pi R^2$ , using the exact identity $\sum_0^L(2\ell+1)=(L+1)^2$ .

Step 3. Elastic column: each absorbed wave contributes $|1-S_\ell|^2=|1-0|^2=1$ — its maximal elastic weight, despite zero phase shift. The sum is identical: $\sigma_{\mathrm{el}}=\frac{\pi}{k^2}(L+1)^2\to\pi R^2$ . Why an absorbed wave still scatters: removing wave $\ell$ leaves a hole in the wavefront, and a wavefront with a hole is not a plane wave — the difference propagates as forward-peaked diffraction. The shadow lobe's angular shape is computed in this lane's Eikonal Approximation article; its total weight is owned here.

Step 4. Total: $\sigma_{\mathrm{tot}}=2\pi R^2$ — twice the geometric area, consistent with the Optical Theorem since the forward amplitude pays for both columns. Convergence is quantitative: $\sigma/(\pi R^2)$ = 1.21, 1.02, 1.002, 1.0002 at $kR=10,10^2,10^3,10^4$ — the excess is the $(L+1)^2$ vs $(kR)^2$ edge effect. Verified: finite sums and convergence table in verify_aq-qm-inelastic-multichannel-scattering.py (section 5); the sum identity is checked exactly for several $L$ .

Common trapAn inelasticity below one does not mean probability is leaking out of the universe

Seeing $|S_\ell|=\eta_\ell<1$ (or a complex phase shift, or an imaginary potential) tempts the conclusion that probability conservation failed. It did not: the full S-matrix is exactly unitary — $\sum_b|S_\ell^{ba}|^2=1$ , verified to $10^{-16}$ in the two-channel solve — and the deficit $1-\eta_\ell^2$ is precisely $\sum_{b\neq a}|S_\ell^{ba}|^2$ , probability sitting in channels you are not watching. The reverse error is as common: assuming a strong absorber has $\sigma_{\mathrm{el}}\approx0$ . Unitarity forces the opposite — the floor $(1-\eta_\ell)^2$ is unavoidable, and a perfect absorber scatters as much as it eats.

Worked examples

Example 1 — cadmium's giant appetite is a sub-permille unitarity deficit. The famous thermal-neutron absorber: $^{113}$ Cd with $\sigma_{\mathrm{abs}}\approx 20{,}600$ b (the benchmark anchored in Differential and Total Cross Section). How close is that to total absorption?

Step 1. Enumerate the budget. A thermal neutron ( $E=25.3$ meV) has $k=3.49\times10^{10}\ \mathrm{m^{-1}}$ ( $\lambda=1.80$ Å); thermal scattering is pure s-wave, so the maximum reaction cross section is $\pi/k^2 = 2.57\times10^{7}$ b — twenty-five million barns. Long wavelengths buy enormous unitarity budgets.

Step 2. Apply the ledger: $1-\eta_0^2 = 20{,}600/2.57\times10^7 = 8.0\times10^{-4}$ , so $\eta_0 = 0.99960$ . Reactor engineering's most notorious absorber spends just 0.08% of the s-wave budget — "giant" thermal cross sections measure the hugeness of $\lambda^2$ , not saturation of the wave.

Step 3. Boundary number — the unavoidable echo: $\sigma_{\mathrm{el}}\ge(\pi/k^2)(1-\eta_0)^2 = 2.57\times10^7\times(4.0\times10^{-4})^2 \approx$ 4.1 b even at $\delta_0=0$ . Cadmium's measured thermal elastic scattering is indeed a few barns — the floor is most of the answer. (For attenuation both columns count: natural Cd at $\sigma\approx2450$ b, $n=4.63\times10^{22}\ \mathrm{cm^{-3}}$ gives the 88 μm mean free path computed in Differential and Total Cross Section — recomputed green here.)

Example 2 — how black is the proton at the LHC? At $\sqrt s=13$ TeV, $pp$ scattering has $\sigma_{\mathrm{tot}}\approx110$ mb and $\sigma_{\mathrm{el}}\approx30$ mb. Test the black-disk hypothesis.

Step 1. Black disk predicts $\sigma_{\mathrm{el}}/\sigma_{\mathrm{tot}}=1/2$ . Measured: $30/110=0.273$ — the proton is grey, not black.

Step 2. Model it as a uniform grey disk: $\eta$ constant for $b<R$ , $\delta=0$ . The ledger gives per-area entries $\sigma_{\mathrm{el}}=\pi R^2(1-\eta)^2$ , $\sigma_{\mathrm{inel}}=\pi R^2(1-\eta^2)$ , $\sigma_{\mathrm{tot}}=2\pi R^2(1-\eta)$ , hence $\sigma_{\mathrm{el}}/\sigma_{\mathrm{tot}}=(1-\eta)/2=0.273$ , so $\eta\approx$ 0.45.

Step 3. Extract the size: $R=\sqrt{\sigma_{\mathrm{tot}}/[2\pi(1-\eta)]}=\sqrt{11\ \mathrm{fm^2}/(2\pi\times0.546)}\approx$ 1.79 fm — well beyond the 0.84-fm charge radius: the interaction region outgrows the charge distribution. (A fully-black fit would give 1.32 fm and overpredict $\sigma_{\mathrm{el}}$ by 80%.) All numbers verified.

Origins · 1 minHerman Feshbach and how this was discovered

Herman Feshbach

1917–2000 · New York, MIT

Herman Feshbach spent a career insisting that nuclear reactions — the messiest processes in low-energy physics — obey exact, elegant bookkeeping. Born in New York City in 1917, he joined MIT in 1941 and stayed six decades. With Porter and Weisskopf he built the 1954 "cloudy crystal ball": a complex optical potential whose imaginary part summarizes all untracked channels, turning neutron–nucleus data into two fittable functions. In 1958 and 1962 he constructed the unified theory of nuclear reactions: project the Schrödinger equation onto open and closed subspaces, and resonances emerge as closed-space bound states leaking through the coupling — the mechanism now bearing his name. He co-authored standard texts, led MIT's physics department and the American Physical Society, and died in 2000 — just as Zeeman-steered Feshbach resonances were becoming the interaction dial of every ultracold quantum gas since.

History

1936

— Bohr's compound-nucleus picture: the projectile is captured and shared among many nucleons before anything re-emerges

1954

— Feshbach, Porter and Weisskopf publish the optical model of neutron scattering: a complex potential with fitted absorption reproduces cross sections across the periodic table

1958

— Feshbach's unified theory of nuclear reactions, Part I: open/closed projection operators and the resonance mechanism

1962

— Unified theory Part II: overlapping resonances, doorway states, and the full multichannel machinery

1998

— Inouye and co-workers observe a magnetically tuned Feshbach resonance in a sodium Bose–Einstein condensate, turning the closed channel into a laboratory dial

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, Ch. 6 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 2, Ch. VIII complements · Newton, Scattering Theory of Waves and Particles, Chs. 16–17 · Taylor, Scattering Theory, Chs. 16–17 · Feshbach, Ann. Phys. 5, 357 (1958); 19, 287 (1962) · Chin, Grimm, Julienne & Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).

Provenance. All cross-section formulas, the inelasticity parametrization $S_\ell=\eta_\ell e^{2i\delta_\ell}$ , the generalized optical theorem, and the black-disk result ( $\sigma_{el}=\sigma_{reac}=\pi R^2$ , $\sigma_{tot}=2\pi R^2$ ) follow Sakurai & Napolitano Modern QM Ch. 6 and Newton Scattering Theory Chs. 16-17. A verifier should: (1) confirm the algebra of the optical-theorem derivation by taking Im of the forward amplitude; (2) check the black-disk partial-wave sums using $\sum_0^L(2\ell+1)=(L+1)^2$ with $L=kR$ ; (3) verify column normalization of a unitary matrix gives $|S^{aa}|\le1$ ; (4) confirm the continuity-equation sink term $-2W|\psi|^2/\hbar$ from $U=V-iW$ . No nonstandard numbers are quoted. — Base note: the exported base carried a stray mid-derivation self-correction ("wait, careful…") inside its optical-theorem step; the algebra it second-guessed was in fact right, and is verified symbolically here. Every base formula checked green; no physics errors found.

Builds on: The S-Matrix and Unitarity in Scattering · Partial-Wave Expansion · Phase Shifts · Optical Theorem · Lippmann-Schwinger Equation. Leads to: Optical Theorem · Scattering by a Screened Coulomb Potential · Differential and Total Cross Section.

If the full S-matrix is unitary, why do we bother with the non-unitary S_ℓ at all?

Because experiments rarely resolve every exit channel. A detector behind a cadmium foil sees the elastic channel; the gamma-cascade channels are invisible. The single number $\eta_\ell e^{2i\delta_\ell}$ is what the elastic-only experiment measures, and the partition theorem converts its modulus deficit into the summed strength of everything unseen — a compression, not an approximation: $1-\eta_\ell^2=\sum_{b\ne a}|S^{ba}_\ell|^2$ identically.

Is the optical potential a fundamental object or a fit?

A fit with a pedigree. Feshbach's projection formalism shows that eliminating untracked channels exactly produces an effective potential — but that exact object is nonlocal, energy-dependent, and as hard as the original problem. The practical optical model truncates it to a local complex well. The imaginary part is genuinely forced (flux does leave); its Woods–Saxon shape is phenomenology that works remarkably well.

How can absorbing a wave produce elastic scattering — where does the scattered wave physically come from?

From the hole. Remove part of a plane wavefront and what continues is "plane wave minus a disk-shaped bite"; the bite spreads by diffraction — Babinet's principle. The scattered wave is not emitted by the absorber; it is the beam's own amplitude rearranging around the missing piece. That is why $\delta_\ell$ can be zero with the elastic term maximal: shadow scattering needs no phase rotation, only a bite.

Do closed channels have any observable effect if they carry no flux?

Yes — the most dramatic effects in the subject. A closed channel modifies the wavefunction at finite $r$ , shifting phase shifts; and if it hosts a quasi-bound state near the open threshold, the open channel acquires a Feshbach resonance tracking that state. Ultracold interaction tuning is nothing but Zeeman-steering a closed channel you can never directly populate.

Why does the reaction cross section blow up as 1/v at low energy — doesn't a slow projectile interact less?

It interacts longer. Wigner's threshold analysis: for an exothermic s-wave process $1-\eta_0^2$ scales as $k$ while the unitarity weight $\pi/k^2$ grows as $1/k^2$ ; the product goes as $1/k\propto1/v$ . Physically the capture matrix element is energy-insensitive (the exit channel is fast regardless), so the rate per unit flux inflates with the dwell time.

Does a complex phase shift mean the same thing as η < 1?

Identical bookkeeping. Write $S_\ell=e^{2i\delta_\ell}$ with $\delta_\ell=\delta_R+i\delta_I$ : then $|S_\ell|=e^{-2\delta_I}\equiv\eta_\ell$ — a positive imaginary phase shift is absorption. The $\eta,\delta$ split separates measurable modulus from measurable phase; older nuclear literature runs on complex $\delta_\ell$ throughout.