Scattering Theory
Bound states are the exception. Most of what is known about the microscopic world was learned by throwing something at something else and counting what came out at each angle, and scattering theory is the dictionary between that count and the interaction responsible for it. The subject begins with Rutherford's analysis of large-angle alpha deflection [Rutherford:1911], which turned a measured angular distribution into the nuclear atom; it acquired its quantum form with Born's collision papers [Born:1926a] [Born:1926b], in which the probability interpretation of The Postulates of Quantum Mechanics and the approximation scheme that still bears his name were introduced in the same breath; and it supplies the language — cross section, phase shift, resonance width, S-matrix — in which nuclear physics (Nuclear Forces and Nuclear Structure), particle physics (Experiment: Deep Inelastic Scattering) and cold-atom physics (Experiment: Bose–Einstein Condensation) all report their results.
The chapter is placed after the approximation methods of Approximation Methods, whose time-dependent perturbation theory it uses, and after identical particles (Identical Particles), because the interference between direct and exchange amplitudes is measurable and was measured [Mott:1930]. It is deliberately nonrelativistic and potential-based; the relativistic S-matrix, its analyticity and its diagrammatic evaluation belong to Canonical Quantization of Fields and Quantum Electrodynamics and Renormalization, and the formal structure stated here is what those chapters build on. The standard monographs are [Taylor:1972] [Newton:1982]; the elementary treatments are [Sakurai:2017] [Griffiths:2018].
Scattering Theory: all derivations of this chapter are pending.
What a scattering experiment measures
Cross sections
[Reserved: the differential cross section defined as scattered flux per unit solid angle divided by incident flux per unit area, so that it carries the dimensions of an area per unit solid angle and is quoted in \(\mathrm{m}^{2}\) per steradian; the integrated and total cross sections; the practical unit of nuclear and particle physics, the barn, defined as exactly \(10^{-28}\,\mathrm{m}^{2}\) and quoted in this treatise with that conversion given once; luminosity and event rate, connecting the theoretical quantity to what a detector counts; the transport and momentum-transfer cross sections, which are the ones that enter the kinetic theory of Kinetic Theory of Gases and the transport coefficients of Nonequilibrium Thermodynamics and Transport.]
Kinematics, frames and flux
[Reserved: the reduction of the two-body problem to one body of reduced mass in a fixed potential, as in Central Forces and Statics; the laboratory-to-centre-of-mass transformation of angles and of the cross section, with the Jacobian written out, since almost every experimental table needs it; elastic, inelastic and reactive channels; the energy variables and the threshold condition; a remark that the relativistic kinematics of Relativistic Dynamics replaces this section wherever the beam energy demands it.]
The scattering amplitude
[Reserved: the asymptotic form \(\psi\sim\ee^{\ii k z}+f(\theta,\phi)\,\ee^{\ii k r}/r\) and the identification \(\dd\sigma/\dd\Omega=\abs{f}^{2}\) by computing the probability current of The Postulates of Quantum Mechanics; that the amplitude is complex while the cross section is not, so that a measured angular distribution determines \(\abs{f}\) and leaves the phase to be inferred — the phase problem that recurs throughout the chapter, and the reason interference experiments such as Section 86.6.2 carry information that a rate measurement cannot.]
The stationary formulation
Asymptotic conditions and scattering states
[Reserved: what it means for an interacting state to approach a free state in the far past and far future; the Møller wave operators \(\Omega_{\pm}\) and their isometry [Moller:1945]; asymptotic completeness — that the ranges of the wave operators together with the bound states exhaust the Hilbert space — stated as the theorem it is and not assumed, with the potentials for which it is proved; the requirement of a potential falling faster than \(1/r\), which the Coulomb case violates and which is why Section 86.6 is treated separately.]
Green functions and the Lippmann–Schwinger equation
[Reserved: the outgoing free Green function \(-\ee^{\ii k\abs{\vect{x}-\vect{x}'}}/4\pi\abs{\vect{x}-\vect{x}'}\) and the \(\ii\varepsilon\) prescription that selects it, with the contour integration of Complex Analysis; the integral equation of Lippmann and Schwinger [Lippmann:1950] in both wavefunction and operator form, and the transition operator \(T\) defined by it; the equivalence to the differential equation with a boundary condition, and why the integral form is the useful one; the formal theory of Gell-Mann and Goldberger [GellMann:1953a] relating the stationary and time-dependent pictures, and Dirac's golden rule [Dirac:1927] as the first-order rate that Approximation Methods supplies.]
The Born series and the Born approximation
[Reserved: iteration of the Lippmann–Schwinger equation; the first Born approximation, in which the amplitude is the Fourier transform of the potential in the momentum transfer \(\vect{q}\) [Born:1926a] [Born:1926b]; the immediate consequences — that a measured angular distribution at high energy is a direct image of the potential, which is the logic of form factors and of Experiment: Deep Inelastic Scattering; the worked cases, Yukawa and screened Coulomb, with the screened Coulomb limit reproducing the Rutherford formula exactly; the validity criterion, the convergence of the series, and its failure at low energy and for strong potentials, where Section 86.3 takes over.]
Partial waves and phase shifts
The partial-wave expansion
[Reserved: expansion of the plane wave and the amplitude in Legendre polynomials for a central potential, with the special functions of Ordinary Differential Equations and Sturm–Liouville Theory; the radial equation, the centrifugal barrier, and the phase shift \(\delta_{\ell}\) defined by the asymptotic form of the radial function; the amplitude \(f(\theta)=k^{-1}\sum_{\ell}(2\ell+1)\ee^{\ii\delta_{\ell}} \sin\delta_{\ell}P_{\ell}(\cos\theta)\); the systematic use of this expansion for slow electrons in gases by Faxén and Holtsmark [Faxen:1927]; the sign of the phase shift as the sign of the potential, and the semiclassical estimate \(\ell_{\max}\approx kR\) that says how many partial waves a given energy needs.]
Unitarity and the optical theorem
[Reserved: conservation of probability as unitarity of the partial wave amplitudes, \(\abs{S_{\ell}}=1\) for purely elastic scattering and \(\abs{S_{\ell}}<1\) when channels are open; the optical theorem \(\sigma_{\text{tot}}=(4\pi/k)\operatorname{Im}f(0)\), derived for potential scattering by Feenberg [Feenberg:1932] and stated in the general form covering inelastic channels by Bohr, Peierls and Placzek [Bohr:1939a]; the physical reading, that the total removal of flux from the beam is the shadow cast by forward interference; the unitarity bound \(\sigma_{\ell}\le4\pi(2\ell+1)/k^{2}\) and its use as a consistency check on measured cross sections.]
Low energy: scattering length and effective range
[Reserved: the dominance of the \(s\) wave as \(k\to0\), the scattering length \(a\) defined by \(\delta_{0}\to-ka\), and the low-energy cross section \(4\pi a^{2}\); the effective-range expansion \(k\cot\delta_{0}=-1/a+\tfrac{1}{2}r_{0}k^{2}+\cdots\), whose model-independence Bethe established [Bethe:1949]; the neutron–proton system as the worked case, with the triplet and singlet scattering lengths of about \(5.4\,\mathrm{fm}\) and \(-23.7\,\mathrm{fm}\), the deuteron as the near-threshold bound state the first implies and the virtual state the second implies, and the resulting hugely spin-dependent cross section — a number measured before it was understood (Nuclear Forces and Nuclear Structure); the Feshbach-resonance tuning of \(a\) in cold gases [Inouye:1998] [Chin:2010], which is what makes Experiment: Bose–Einstein Condensation and Superconductivity and Superfluidity quantitative.]
As the collision energy is lowered, elastic scattering becomes isotropic — the measured angular distribution flattens until no direction is preferred — and the total elastic cross section approaches a constant,
fixed by a single length \(a\) characteristic of the colliding pair. Two potentials of quite different shape that happen to share the value of \(a\) cannot be told apart by any low-energy measurement. The length need not resemble the range of the force: it may be many times larger, and it may be negative. Both extremes are observed. The neutron–proton system has scattering lengths of opposite sign and very different magnitude in its two spin channels, which is why the low-energy neutron cross section on hydrogen depends so violently on the relative spin orientation; and in a cold atomic gas \(a\) is swept through zero and through infinity by an applied magnetic field tuned across a Feshbach resonance [Inouye:1998], so that the same atoms behave as an ideal gas or as a strongly interacting fluid on demand.
Derivation. Let the potential have finite range \(R\). For \(r>R\) the radial function of angular momentum \(\ell\) is a combination of free solutions, and its amplitude inside the range is suppressed by the centrifugal barrier: the regular free solution behaves as \((kr)^{\ell+1}\) near the origin, so the potential is sampled with weight \((kR)^{2\ell+1}\) and
Only \(\ell=0\) survives the limit. Defining \(a:=-\lim_{k\to0}\delta_{0}(k)/k\), the partial-wave amplitude of Section 86.3.1 gives
which is independent of \(\theta\): the scattering is isotropic, and \(\sigma=\int\abs{f}^{2}\dd\Omega=4\pi a^{2}\), which is Equation (86.1). The geometric meaning of \(a\) follows from the same limit. Outside the range the \(s\)-wave radial function is \(u\propto\sin(kr+\delta_{0})\to k\,(r-a)\), so \(a\) is the intercept at which the exterior wave function, extrapolated inwards as a straight line, crosses zero. Everything the interior does to the collision is compressed into that one intercept, and a large \(\abs{a}\) signals a bound or virtual state lying close to threshold rather than a large force.
∎Levinson's theorem
[Reserved: the relation \(\delta_{\ell}(0)-\delta_{\ell}(\infty) =n_{\ell}\pi\) between the total change of a phase shift and the number of bound states of that angular momentum [Levinson:1949], with the half-integer modification for a zero-energy resonance in the \(s\) wave; the statement that a scattering measurement therefore counts bound states it cannot produce; the inverse-scattering problem, in which the potential is reconstructed from the phase shifts and the bound-state data, solved by Gel'fand and Levitan [Gelfand:1951], together with the non-uniqueness that phase-equivalent potentials exhibit.]
The S-matrix
The S operator and its symmetries
[Reserved: \(S=\Omega_{-}^{\dagger}\Omega_{+}\) as the map from incoming to outgoing asymptotic states, with the name and the resonating-group construction due to Wheeler [Wheeler:1937] and its elevation to the primary object of the theory — observable quantities only — by Heisenberg [Heisenberg:1943]; unitarity from probability conservation, symmetry from time reversal and the resulting reciprocity or detailed-balance relation between forward and reverse reactions, and the constraints from rotational invariance, parity and the discrete symmetries of Discrete Symmetries and CPT; the observation that this is where Axiomatic Quantum Field Theory begins.]
Analyticity, poles and dispersion relations
[Reserved: continuation of the partial-wave amplitude to complex energy or complex momentum; poles on the positive imaginary \(k\) axis as bound states, poles on the negative imaginary axis as virtual states, and poles on the second sheet as resonances; the branch cuts and the physical sheet; dispersion relations as Cauchy's theorem (Complex Analysis) applied with causality, in the form given for light by Kronig [Kronig:1926] and Kramers [Kramers:1927] and transplanted to scattering amplitudes; complex angular momentum and Regge trajectories [Regge:1959], named here because the linear mass-squared–spin trajectories of the observed hadron spectrum are their evidence.]
Resonances
The Breit–Wigner form
[Reserved: the resonant partial-wave amplitude \(\Gamma/2\) over \((E_{R}-E)-\ii\Gamma/2\), giving a Lorentzian peak in the cross section of height set by the unitarity bound and width \(\Gamma\), first written for slow-neutron capture by Breit and Wigner [Breit:1936]; the mean life \(\tau=\hbar/\Gamma\) and the phase shift sweeping through \(\pi/2\); partial widths and branching ratios; the worked examples — neutron resonances in heavy nuclei (Nuclear Forces and Nuclear Structure), the \(\Delta\) resonance in pion–nucleon scattering, and the \(Z\) line shape whose measured width counts the light neutrino species (Electroweak Unification and the Higgs Boson).]
Cross sections are not smooth functions of energy. Superposed on a slowly varying background are peaks whose shape is, to good accuracy, a Lorentzian of full width \(\Gamma\) at half maximum centred on an energy \(E_{R}\). Slow neutrons incident on a heavy nucleus show hundreds of them within the first few electronvolts; pion–nucleon scattering shows one dominant peak near the first excited state of the nucleon; and the cross section for electron–positron annihilation into hadrons shows the \(Z\). Wherever the corresponding unstable state can also be timed independently, its mean life is found to be \(\tau=\hbar/\Gamma\), and the widths tabulated for hundreds of such states [Navas:2024] are read as lifetimes on exactly that understanding. A resonance is therefore not a new kind of object but a state that lives long enough to be counted and too briefly to be caught.
Derivation. Consider a state that, once formed, decays exponentially: its amplitude is \(a(t)=\ee^{-\ii E_{R}t/\hbar}\ee^{-\Gamma t/2\hbar}\) for \(t>0\) and zero before, so that \(\abs{a(t)}^{2}=\ee^{-\Gamma t/\hbar}\) and the mean life is \(\tau=\hbar/\Gamma\). Its energy content is the Fourier transform of Fourier Analysis and Integral Transforms,
the integral converging because of the damping factor. Hence the probability of finding the state at energy \(E\) is
a Lorentzian whose full width at half maximum is exactly \(\Gamma\). The cross section for forming the state through a given entrance channel is proportional to this same distribution, so the observed line shape and the observed lifetime are two readings of one complex pole at \(E=E_{R}-\ii\Gamma/2\); the relation \(\tau\Gamma=\hbar\) between them is not an approximation but the statement that the two are Fourier conjugates, the time–energy form of the uncertainty relation of The Postulates of Quantum Mechanics. In partial-wave language the same pole appears as an elastic amplitude proportional to \((\Gamma/2)\bigl[(E_{R}-E)-\ii\Gamma/2\bigr]^{-1}\), whose modulus attains the unitarity bound of Section 86.3.2 at \(E=E_{R}\) and whose phase shift sweeps through \(\pi/2\) there — which is how a resonance is distinguished from a mere bump in the background.
∎Time delay
[Reserved: the Eisenbud–Wigner delay \(2\hbar\,\dd\delta_{\ell}/\dd E\) as the time a wave packet lingers in the interaction region, with Wigner's lower bound on it — causality forbids arbitrarily large negative delay, that is arbitrarily early emission [Wigner:1955]; Smith's lifetime matrix generalizing this to many channels [Smith:1960]; the equivalence of a long delay and a narrow resonance; the modern measurement of photoionization delays on the attosecond scale as the direct observation of this quantity.]
Fano profiles and Feshbach resonances
[Reserved: interference between a resonant and a direct amplitude producing the asymmetric line shape parametrized by Fano's \(q\) [Fano:1961], with autoionizing states of helium as the original observation; Feshbach's unified theory of nuclear reactions, in which a closed channel supports a quasi-bound state coupled to an open one [Feshbach:1958b]; the optical model, in which the average effect of many open channels is a complex potential [Feshbach:1954], and the absorption cross section it predicts; the same physics used as a control knob in ultracold gases [Inouye:1998] [Chin:2010].]
Coulomb scattering
Rutherford, classically and quantum mechanically
[Reserved: the classical derivation of the \(\csc^{4}(\theta/2)\) law from the hyperbolic orbits of Central Forces and Statics [Rutherford:1911], and the Geiger–Marsden measurements that tested every one of its dependences [Geiger:1913]; the exact quantum solution in parabolic coordinates by Gordon [Gordon:1928b], giving the same cross section but a logarithmically distorted phase and a modified asymptotic form, so that the long range of the Coulomb potential invalidates Section 86.2.1 while leaving the measured rate unchanged — a coincidence of the inverse-square law that deserves to be flagged rather than passed over; the Mott cross section, which adds the spin of a relativistic projectile [Mott:1929] and whose deviation from Rutherford at large angle is a measurement of nuclear size.]
A beam of alpha particles crossing a thin metal foil is deflected, and a small but perfectly reproducible fraction is deflected through more than a right angle — some of it back towards the source [Geiger:1909]. The number counted per unit solid angle at a scattering angle \(\theta\) obeys
with \(E\) the kinetic energy in the centre-of-mass frame. Four dependences are asserted here and all four were measured separately: the fourth power of the sine of the half angle, the inverse square of the energy, the square of the charge number of the target, and — for a foil thin enough that a particle meets at most one scatterer — strict proportionality to the foil thickness [Geiger:1913]. Together they establish that the positive charge of an atom is not spread through it but concentrated in a region small enough to permit the close approaches that large angles require.
Derivation. Let the projectile of charge \(Z_{1}e\) approach a fixed charge \(Z_{2}e\) with reduced mass \(\mu\), asymptotic speed \(v\), energy \(E=\tfrac{1}{2}\mu v^{2}\) and impact parameter \(b\), and write \(\kappa=Z_{1}Z_{2}e^{2}/4\pi\epsilon_{0}\) so that the potential is \(\kappa/r\). The orbit is the repulsive hyperbola of Central Forces and Statics, whose asymptotes make the deflection angle \(\theta\) given by
The map \(b\mapsto\theta\) is monotone, so particles entering the annulus between \(b\) and \(b+\dd b\) leave between \(\theta\) and \(\theta+\dd\theta\), and equating the two fluxes, \(2\pi b\,\abs{\dd b}=(\dd\sigma/\dd\Omega)\,2\pi\sin\theta\, \abs{\dd\theta}\), gives the classical cross-section formula
Differentiating Equation (86.6) gives \(\abs{\dd b/\dd\theta}=(\kappa/4E)\sin^{-2}(\theta/2)\), while \(\sin\theta=2\sin(\theta/2)\cos(\theta/2)\) turns the first factor into \(b/\sin\theta=(\kappa/4E)\sin^{-2}(\theta/2)\) as well. Their product is
which is Equation (86.5). The thickness dependence follows from counting: with \(n\) scatterers per unit volume in a foil of thickness \(t\), the fraction of the beam deflected into \(\dd\Omega\) is \(nt\,(\dd\sigma/\dd\Omega)\dd\Omega\), linear in \(t\) so long as \(nt\sigma\ll1\).
Two remarks belong with the result rather than after it. First, nothing in the derivation refers to \(\hbar\), and yet the first Born approximation of Section 86.2.3 and the exact quantum solution in parabolic coordinates both return Equation (86.5) unchanged. That agreement is a peculiarity of the inverse-square law and must not be read as a general licence to compute cross sections classically. Second, the total cross section obtained by integrating Equation (86.5) diverges, because \(\sin^{-4}(\theta/2)\) is not integrable at \(\theta=0\): an unscreened Coulomb potential deflects every particle, however distant. Real targets are screened by their own electrons, which cuts the divergence off at an angle of order the inverse of the screening length times the momentum, and it is that screening — not the formula — that makes a total cross section finite.
∎Identical particles: Mott scattering
[Reserved: when projectile and target are identical, the direct and exchange amplitudes must be added with the exchange parity of Identical Particles before squaring, giving an interference term with no classical analogue and a cross section symmetric about \(90\) degrees in the centre-of-mass frame; Mott's calculation for two electrons [Mott:1930] — customarily cited to his 1929 paper, which is a different result — and its confirmation in alpha-on-helium scattering [Chadwick:1930], where the interference maximum is a factor of two above the classical sum; the spin-averaged version, weighting singlet and triplet, as the cleanest laboratory demonstration that exchange symmetry is a measurable property of nature and not a bookkeeping convention.]
When projectile and target are identical, the measured centre-of-mass angular distribution is symmetric about \(90^\circ\) — as it must be, since the two particles arriving at the detectors cannot be told apart — and it is not the sum of the two contributions that a trajectory picture would add. For identical spinless particles, such as an alpha particle scattering on helium, the counting rate at exactly \(90^\circ\) is twice the classical sum, with oscillations on either side of it [Chadwick:1930]. For unpolarized identical spin-half particles the interference enters with the opposite sign in the triplet, which has three times the weight of the singlet, and the measured rate at \(90^\circ\) is instead half the classical sum. The sign of a term in a cross section is thus a directly measurable consequence of the exchange symmetry postulated in Identical Particles.
Derivation. Detecting a particle at angle \(\theta\) in the centre-of-mass frame can happen in two ways: the projectile was deflected by \(\theta\), amplitude \(f(\theta)\), or the projectile was deflected by \(\pi-\theta\) and the recoiling target reached the detector, amplitude \(f(\pi-\theta)\). The two possibilities end in the same final configuration and are indistinguishable in principle, so the amplitudes are added before squaring, with the relative sign fixed by the required symmetry of the spatial state:
the upper sign for a symmetric spatial state — identical spinless bosons, or spin-half particles in the spin singlet — and the lower sign for an antisymmetric one. The classical prediction is the first two terms alone; the third has no classical counterpart, and both expressions are manifestly symmetric under \(\theta\mapsto\pi-\theta\).
At \(\theta=\pi/2\) the two amplitudes coincide, \(f(\theta)=f(\pi-\theta)\), and the arithmetic is immediate. The symmetric case gives \(\abs{2f}^{2}=4\abs{f}^{2}\) against a classical \(2\abs{f}^{2}\): a factor of two. The antisymmetric case gives exactly zero, however strong the interaction. For unpolarized spin-half particles the four spin states are equally likely, so the singlet carries weight \(\tfrac{1}{4}\) and the triplet \(\tfrac{3}{4}\); since a symmetric spin state (triplet) forces an antisymmetric spatial state and conversely, the spin-averaged rate at \(90^\circ\) is
one half of the classical \(2\abs{f}^{2}\). The measured ratio at \(90^\circ\) therefore distinguishes bosons from fermions with no adjustable parameter, and it does so through the value of the cross section and not through any spin-dependent force.
∎Observed effects the formalism must reproduce
The Ramsauer–Townsend effect
[Reserved: the near-vanishing of the cross section for slow electrons in argon, krypton and xenon near \(0.7\,\mathrm{eV}\), reported independently by Ramsauer [Ramsauer:1921] and by Townsend and Bailey [Townsend:1922]; its explanation as an \(s\)-wave phase shift passing through \(\pi\), so that the leading partial wave contributes nothing — a purely quantum result with no classical counterpart, and historically one of the earliest; the corresponding transparency window and its consequence for electron transport in noble gases; the analogous zero-crossing of the scattering length used to make non-interacting Fermi gases.]
Threshold behaviour
[Reserved: Wigner's threshold laws, giving the energy dependence of a cross section just above the opening of a channel as a power fixed by the angular momentum and the range of the interaction [Wigner:1948]; the \(1/v\) law for exothermic reactions on which slow-neutron absorption and the operation of every thermal reactor rest (Nuclear Forces and Nuclear Structure); cusps at thresholds as a unitarity consequence rather than a dynamical one; the modification for long-range potentials.]
For an exothermic reaction — one whose outgoing channel is open with a large energy release, radiative neutron capture being the standard case — the cross section measured at low incident energy rises as the inverse of the incident speed over several decades, so that the product \(\sigma v\) is constant. Absorption of slow neutrons in boron, in cadmium and in most heavy nuclei follows this law from thermal energies downwards, interrupted only by the resonances of Phenomenon 86.2. The law is why a reactor moderates its neutrons before it asks them to be absorbed: slowing a neutron buys cross section in exact proportion to the slowing (Nuclear Forces and Nuclear Structure).
Derivation. At low incident energy only the \(\ell=0\) wave reaches the target, by Equation (86.2), so the reaction cross section is
Write the exterior \(s\)-wave radial function as \(u(r)=\ee^{-\ii kr}-S_{0}\ee^{\ii kr}\) and let \(f=R\,u'(R)/u(R)\) be its logarithmic derivative at the surface \(r=R\) of the interacting region. Matching to the interior solution makes \(f\) complex — its imaginary part is the flux disappearing into the reaction channel — and, crucially, the reaction channel is open by an energy of nuclear size, so \(f\) is insensitive to the tiny incident energy and may be treated as a constant \(f=f_{1}+\ii f_{2}\) with \(f_{2}<0\). Solving the matching condition for \(S_{0}\),
so that
which is positive and, as \(kR\to0\), proportional to \(k\). Substituting into Equation (86.9),
the constant being fixed by the interior alone. The physical content is worth separating from the algebra: the two powers of \(1/k\) in Equation (86.9) are the geometric size of the incident \(s\) wave, which grows without limit as the wavelength does, while the single factor of \(k\) is the small amplitude that wave has at the target. An endothermic reaction behaves quite differently, because there the outgoing momentum vanishes at threshold and supplies its own energy dependence.
∎High energy: the eikonal and Glauber regime
[Reserved: the eikonal approximation, in which the phase accumulated along a straight-line path replaces the full solution — the same short-wavelength limit as the optical–mechanical analogy of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, here applied to the quantum amplitude; the profile function and the impact-parameter representation; Glauber's multiple-scattering theory for composite targets [Glauber:1955], which is how the deuteron and then nuclei are treated as collections of scatterers; the black-disc limit and the factor-of-two total cross section, whose shadow term is the optical theorem of Section 86.3.2 in its most physical form; the pointer to the relativistic treatment in Quantum Electrodynamics and Renormalization and Quantum Chromodynamics.]
At wavelengths short compared with the target, and for a target that absorbs nearly everything striking it, the measured total cross section is not the geometric area \(\pi R^{2}\) but twice it. One half is accounted for by the absorption itself; the other half is elastic scattering confined to a narrow forward cone of angular width of order \(1/kR\) and carrying diffraction minima, from whose spacing \(R\) can be read off. The pattern is standard in fast-neutron scattering from nuclei and in high-energy hadron collisions. It has no classical counterpart whatever: a classical opaque disc removes \(\pi R^{2}\) of flux, and nothing more.
Derivation. Model the target as a black disc: in the partial-wave expansion of Section 86.3.1 set \(S_{\ell}=0\) for \(\ell\leq L\), with \(L=kR\), and \(S_{\ell}=1\) for \(\ell>L\). The two standard sums of Section 86.3.2 then read
using \(\sum_{\ell=0}^{L}(2\ell+1)=(L+1)^{2}\) and \(L=kR\gg1\). Their sum is \(\sigma_{\text{tot}}=2\pi R^{2}\). The same answer follows from the optical theorem: the forward amplitude is \(f(0)=(2\ii k)^{-1}\sum_{\ell}(2\ell+1)(S_{\ell}-1) =\ii(L+1)^{2}/2k\approx\ii kR^{2}/2\), so \(\sigma_{\text{tot}}=(4\pi/k)\operatorname{Im}f(0)=2\pi R^{2}\), in agreement with Equations (86.12) and (86.13).
The physical reading is the one worth keeping. Behind an absorbing target there is a shadow, and a shadow is not an absence of wave but the superposition of the incident wave with a second wave that cancels it there. That cancelling wave must be produced by the target, and it carries as much flux as the absorbed wave removed. Far downstream the shadow wave spreads out of the forward direction — by the diffraction angle \(1/kR\) — and is then counted by a detector at finite angle as elastic scattering. The elastic half of the cross section is thus not an independent process but the necessary accompaniment of absorption, which is why \(\sigma_{\text{el}}\) cannot be made smaller than \(\sigma_{\text{abs}}\) for a fully absorbing target however the interaction is arranged.
∎