Inelastic Scattering Preview

Inelastic scattering is a collision in which the outgoing asymptotic channel has a different internal energy or different channel label from the incoming one. Translational kinetic energy can become rotational, vibrational, electronic, hyperfine, spin, nuclear, or collective excitation energy; the reverse process can release stored internal energy into motion.

The defining distinction is not whether the trajectory bends. Elastic scattering can redirect momentum through a large angle while preserving the internal state. A collision is inelastic when the final asymptotic state carries physically different internal information.

Required background. Cross sections supplies the flux-to-rate relation; the S-matrix describes transitions between asymptotic channels. Fermi’s golden rule is used below for weak coupling.

Elastic, Inelastic, and Reaction Channels

Let an asymptotic channel label $\alpha$ specify the separated objects, their internal quantum numbers, and their relative motion. A transition is written

$$\alpha\longrightarrow\beta.$$

The terminology is convention dependent at the edges, but the following classification is useful.

Process	What changes asymptotically?	Example
elastic	direction or phase, but not the internal channel	ground-state atom scattered through an angle
inelastic excitation	internal state, with the same particle partition	rotational excitation of a molecule
superelastic de-excitation	internal state releases energy into relative motion	collisionally quenched excited atom
rearrangement or reaction	grouping, species, or particle content	charge transfer or chemical reaction
breakup or production	number of asymptotic fragments	molecular dissociation or particle production

In atomic and molecular collision theory, inelastic often means the same constituents in different internal states, while reactive is reserved for a new arrangement. In nuclear and particle physics, inelastic is frequently used more broadly for every nonelastic final channel. A calculation should state which convention its reported inclusive cross section uses.

Internal States and Channel Thresholds

For a projectile and target with relative coordinate $\mathbf r$, a useful free Hamiltonian is

$$H_0 = T_{\mathrm{rel}}+H_{\mathrm{int}}.$$

Let

$$H_{\mathrm{int}} \lvert a\rangle = \epsilon_a\lvert a\rangle.$$

For a fixed two-body partition, the internal energy supplies the channel threshold. More generally, rest energies, binding energies, and the choice of fragments are included in a threshold $E_a^{\mathrm{th}}$. Energy conservation in channel $a$ reads

$$E = E_a^{\mathrm{th}} + \frac{\hbar^2 k_a^2}{2\mu_a},$$

where $\mu_a$ is the channel reduced mass. The channel is open when $E>E_a^{\mathrm{th}}$ and closed when $E<E_a^{\mathrm{th}}$. A closed channel carries no flux to infinity, although virtual coupling to it can still shift an open-channel amplitude or generate a resonance.

A Q-value convention

Define the channel Q-value by

$$Q_{\beta\alpha} \equiv E_\alpha^{\mathrm{th}} - E_\beta^{\mathrm{th}}.$$

Then the final relative kinetic energy is

$$K_\beta = K_\alpha+Q_{\beta\alpha}.$$

With this convention:

• $Q_{\beta\alpha}<0$ is endothermic; the collision must supply internal energy.
• $Q_{\beta\alpha}>0$ is exothermic; internal or binding energy is released into motion.
• $Q_{\beta\alpha}=0$ does not by itself imply an elastic process, because degenerate internal states can still differ.

For an endothermic transition, the channel opens only when

$$K_\alpha \ge -Q_{\beta\alpha}.$$

At fixed total energy $E$, raising the internal threshold from $E_\alpha^{\mathrm{th}}$ to $E_\beta^{\mathrm{th}}$ reduces the outgoing kinetic energy. The exit channel is open only when its threshold lies below $E$. Reversing the thresholds describes an exothermic transition.

Coupling Internal and Relative Motion

Write the interaction as an operator on both relative and internal coordinates,

$$V=V(\mathbf r,\xi),$$

where $\xi$ denotes the target or composite system’s internal variables. Expanding a stationary state in internal eigenstates gives, schematically,

$$\lvert\Psi\rangle = \sum_a \lvert\psi_a\rangle \otimes \lvert a\rangle.$$

The channel-coupling potentials are

$$V_{ba}(\mathbf r) = \langle b\rvert V(\mathbf r,\xi) \lvert a\rangle.$$

Diagonal elements $V_{aa}$ distort motion without changing the internal state. Off-diagonal elements $V_{ba}$ drive transitions. If symmetry forces an off-diagonal matrix element to vanish, that channel is forbidden at the corresponding order even when it is kinematically open.

For a simplified partial-wave problem with one reduced mass, the radial equations have the coupled form

$$\left[ -\frac{\hbar^2}{2\mu} \frac{d^2}{dr^2} + \frac{\hbar^2\ell_b(\ell_b+1)} {2\mu r^2} \right]u_b(r) +(E_b^{\mathrm{th}}-E)u_b(r) +\sum_a V_{ba}(r)u_a(r) =0.$$

This equation makes the mechanism transparent: channel amplitudes exchange probability only where the off-diagonal couplings are appreciable. Full reactive scattering may require different Jacobi coordinates and reduced masses for different arrangements, so one common radial coordinate is then only a schematic guide.

Channel-Resolved Scattering Amplitudes

Choose an incoming channel $\alpha$. In a coordinate-space normalization convenient for cross sections, the large-distance state has the structure

$$\Psi_\alpha^{(+)} \sim{} \Phi_\alpha e^{i\mathbf k_\alpha\cdot\mathbf r} + \sum_{\beta\,\mathrm{open}} \Phi_\beta f_{\beta\alpha}(\Omega_\beta) \frac{e^{ik_\beta r}}{r}.$$

Here $\Phi_a$ is the internal channel state and $f_{\beta\alpha}$ is the amplitude for the resolved transition $\alpha\to\beta$. Different dynamical histories leading to the same final channel contribute coherently to $f_{\beta\alpha}$. Orthogonal final internal channels are distinct records and their probabilities are summed after squaring.

This distinction prevents a common error. If two amplitudes $A_1$ and $A_2$ end in the same state, the probability contains interference:

$$P \propto \lvert A_1+A_2\rvert^2.$$

If they end in orthogonal states $\lvert\beta_1\rangle$ and $\lvert\beta_2\rangle$, an unresolved detector gives

$$P_{\mathrm{inclusive}} \propto \lvert A_{\beta_1}\rvert^2 + \lvert A_{\beta_2}\rvert^2.$$

Cross Sections and the Velocity Factor

The outgoing spherical wave in channel $\beta$ carries radial current proportional to

$$j_{\beta,r} = \frac{v_\beta}{r^2} \lvert f_{\beta\alpha}\rvert^2,$$

while the incident plane wave carries current $j_\alpha=v_\alpha$ in this normalization. Therefore

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega_\beta} = \frac{v_\beta}{v_\alpha} \left| f_{\beta\alpha}(\Omega_\beta) \right|^2,$$

with

$$v_a = \frac{\hbar k_a}{\mu_a}.$$

For the same reduced mass in both channels, the prefactor is $k_\beta/k_\alpha$. It equals one whenever the two speeds agree, including elastic scattering and transitions between degenerate channels with the same reduced mass. The factor is not an optional convention: it converts amplitude into outgoing flux.

If the incoming channel has degeneracy $g_\alpha$ and the experiment does not polarize or resolve internal substates, the reported observable usually averages over initial labels and sums over final labels:

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega_\beta} = \frac{1}{g_\alpha} \frac{v_\beta}{v_\alpha} \times \sum_{m_\alpha,m_\beta} \left| f_{\beta m_\beta,\alpha m_\alpha} \right|^2.$$

The inclusive inelastic cross section is a sum over the channels counted as inelastic by the stated convention:

$$\sigma_{\mathrm{inel},\alpha} = \sum_{\substack{ \beta\,\mathrm{open} \\ \beta\ne\alpha }} \int d\Omega_\beta\, \frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega_\beta}.$$

When the final internal spectrum is continuous, the channel sum becomes an energy integral and one measures distributions such as

$$\frac{d^2\sigma}{d\Omega\,dE_f}.$$

The detector definition decides which quantum numbers are held fixed, summed, averaged, or integrated. Differential and Total Cross Sections gives the general event-rate interpretation.

The Golden-Rule Connection

In weak coupling, a transition from an incoming continuum state $\lvert i\rangle$ to final states near $\lvert f\rangle$ has the golden-rule form

$$d\Gamma_{i\to f} = \frac{2\pi}{\hbar} \left| \langle f\rvert V\lvert i\rangle \right|^2 \delta(E_f-E_i) d\rho_f.$$

For one target, a cross section is the transition rate divided by the incident particle flux density:

$$d\sigma_{i\to f} = \frac{d\Gamma_{i\to f}}{j_{\mathrm{in}}}.$$

With one incoming plane-wave particle normalized in a volume $\mathcal V$,

$$j_{\mathrm{in}} = \frac{v_i}{\mathcal V}.$$

The normalization volume cancels against the continuum density of final states. What remains is the same physical structure seen above: a squared transition amplitude, an energy-conserving phase-space factor, and division by incident flux.

The golden rule is the weak-coupling limit, not the complete scattering theory. Repeated interaction and outgoing boundary conditions are resummed by the transition operator

$$T(E) = V + V G_0^{(+)}(E)T(E).$$

At first Born order, $T\approx V$, and the golden-rule matrix element is recovered. Beyond that order, the exact on-shell $T$-matrix replaces the bare interaction. With the wave-vector-normalized states of T-Matrix, this connection is

$$S_{fi} = \delta_{fi} - 2\pi i\, \delta(E_f-E_i) T_{fi}.$$

Factors of $\hbar$, momenta, and $2\pi$ move when state normalizations change. T-Matrix fixes the nonrelativistic convention used for exact amplitudes, while Fermi’s Golden Rule states the time-scale assumptions behind the rate limit.

A Two-Level Target in First Born Approximation

Consider target states $\lvert g\rangle$ and $\lvert e\rangle$ separated by

$$\epsilon_e-\epsilon_g = \Delta>0.$$

Take an interaction

$$V(\mathbf r,\xi) = V_d(\mathbf r) + W(\mathbf r)\mathcal O(\xi),$$

where $V_d$ is diagonal in the target states and $\mathcal O$ can excite the target. With the Fourier-transform convention

$$\widetilde W(\mathbf q) = \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} W(\mathbf r),$$

the first Born excitation amplitude is

$$f_{eg}^{(\mathrm B)}(\mathbf q) = -\frac{\mu}{2\pi\hbar^2} \widetilde W(\mathbf q) \langle e\rvert\mathcal O\lvert g\rangle.$$

For equal entrance and exit reduced masses,

$$k_e = \sqrt{ k_g^2 - \frac{2\mu\Delta}{\hbar^2} }.$$

The channel is closed when the expression under the square root is negative. Above threshold,

$$\frac{d\sigma_{e\leftarrow g}^{(\mathrm B)}}{d\Omega} = \frac{k_e}{k_g} \left| f_{eg}^{(\mathrm B)}(\mathbf q) \right|^2.$$

Here $\mathbf q=\mathbf k_e-\mathbf k_g$ is the wave-vector transfer and $\hbar\mathbf q$ the projectile momentum change. The unequal speeds mean that its magnitude is

$$q^2 = k_g^2+k_e^2 - 2k_gk_e\cos\theta.$$

This elementary model separates three ingredients:

1. Kinematics: $k_e$ and the channel threshold.
2. Spatial resolution: the form factor $\widetilde W(\mathbf q)$.
3. Internal dynamics: the matrix element $\langle e\rvert\mathcal O\lvert g\rangle$.

A selection rule can make the third factor vanish. A rapidly decaying form factor can suppress large momentum transfer even when the transition is symmetry allowed. A channel can therefore be kinematically open yet dynamically weak.

Threshold Laws

Opening a channel creates nonanalytic energy dependence because

$$k_\beta \propto \sqrt{E-E_\beta^{\mathrm{th}}}.$$

For a short-range interaction and a fixed entrance momentum, the Wigner threshold law for an exit partial wave $\ell_\beta$ has the form

$$\sigma_{\beta\leftarrow\alpha}^{(\ell_\beta)} \propto k_\beta^{2\ell_\beta+1}.$$

Thus an $s$-wave exit channel turns on as $k_\beta$, whereas a $p$-wave exit channel turns on as $k_\beta^3$. The centrifugal barrier suppresses higher exit angular momenta near threshold.

This compact law has qualifications:

• if the entrance channel is simultaneously near its own threshold, entrance powers also matter;
• Coulomb or other long-range tails modify the short-range law;
• a nearby bound state or resonance can dominate the smooth threshold scaling;
• detector resolution can smear the mathematical cusp.

For an exothermic $s$-wave process as $k_\alpha\to0$, the rate coefficient

$$K_{\mathrm{loss}} = v_\alpha\sigma_{\mathrm{inel}}$$

can approach a finite limit even while $\sigma_{\mathrm{inel}}\propto1/k_\alpha$. A divergent cross section does not then imply a divergent event rate per incident density; the incident speed vanishes at the same time.

Conservation Laws and Selection Rules

Energy conservation decides whether a channel is open. Symmetry decides whether it can couple. Depending on the Hamiltonian, useful conserved quantities include total angular momentum, its projection, parity, total spin, exchange symmetry, and charges.

For example, if the coupling operator is a spherical tensor of rank $\lambda$, angular-momentum addition constrains which internal rotational states can be connected. Parity gives an independent condition. The spatial relative-motion partial wave can exchange angular momentum with the target, so a rule for the target alone is not generally a rule for the complete collision.

The reliable procedure is to apply symmetry to the complete channel state and interaction matrix element,

$$\langle\beta\rvert V\lvert\alpha\rangle,$$

not to infer a transition from energy conservation alone. Selection Rules in Transition Rates develops the matrix-element logic, and Identical-Particle Scattering treats exchange constraints when the outgoing particles are indistinguishable.

Exclusive and Inclusive Measurements

An exclusive measurement specifies the final channel as fully as the apparatus permits: internal level, spin state, emission direction, energy, or fragment identity. An inclusive measurement sums over some or all unobserved final channels.

For target response, use $\mathbf Q=\mathbf k_i-\mathbf k_f=-\mathbf q$: the target receives momentum $\hbar\mathbf Q$ and energy $\mathcal E=E_i-E_f=\hbar\omega$. Consider a weak probe coupled to the number density of $N$ identical scatterers in a target with a nondegenerate ground state $|0\rangle$. To match the convention of Structure Factors, define

$$\rho_{\mathbf Q}=\sum_{j=1}^N e^{-i\mathbf Q\cdot\mathbf r_j}, \qquad S_E(\mathbf Q,\mathcal E) =\frac{1}{N}\sum_n \left|\langle n|\rho_{-\mathbf Q}|0\rangle\right|^2 \delta(\mathcal E-E_n+E_0).$$

The minus sign on the operator label matters: $e^{+i\mathbf Q\cdot\mathbf r}|\mathbf p\rangle=|\mathbf p+\hbar\mathbf Q\rangle$, so $\rho_{-\mathbf Q}$ delivers the stated target momentum. $S_E$ is an energy-resolved spectrum per particle, with units of inverse energy. At first Born order, for unchanged projectile mass, the measured spectrum has the schematic form

$$\frac{d^2\sigma}{d\Omega\,dE_f} \propto \frac{k_f}{k_i}\,N S_E(\mathbf Q,E_i-E_f),$$

where the omitted factor contains the probe coupling and its momentum dependence. At fixed incident energy, $|d\mathcal E|=|dE_f|$, so changing between outgoing energy and target energy transfer introduces no additional magnitude factor. In contrast, the angular-frequency spectrum used in the many-body convention satisfies

$$S_\omega(\mathbf Q,\omega)=\hbar S_E(\mathbf Q,\hbar\omega), \qquad \int d\omega\,S_\omega=\int d\mathcal E\,S_E.$$

The $n=0$ term is an elastic line at zero energy transfer; omit it when reporting only inelastic events. This is golden-rule physics written as a target correlation spectrum, with the channel sum replaced by a sum over many-body excitations. Structure Factors develops detailed balance, moments, thermal ensembles, and probe-specific forward models; Linear Response Preview develops the response viewpoint.

Inclusive sums are also what make unitarity operational. If every open final channel is included, lost elastic probability reappears in inelastic or reactive channels. Observing only one elastic element can give

$$\left|S_{\alpha\alpha}^{(\ell)}\right|<1$$

without violating unitarity of the full $S$-matrix. Unitarity derives the channel sum, and Optical Theorem relates the inclusive total cross section to forward elastic scattering.

Detailed Balance

Time-reversal invariance relates a transition to its inverse. For angle-integrated two-body cross sections, with initial substates averaged and final substates summed, a common detailed-balance relation is

$$g_\alpha k_\alpha^2 \sigma_{\beta\leftarrow\alpha} = g_\beta k_\beta^2 \sigma_{\alpha\leftarrow\beta}.$$

Here $g_a$ is the channel degeneracy. The two sides are evaluated at the same total energy and with reversed kinematics. External magnetic fields, polarization conventions, identical-particle factors, or time-reversal-violating dynamics require corresponding care. Detailed balance is a strong normalization and phase-space check because it tests both matrix-element reciprocity and channel counting.

Bridge to Relativistic Reactions

The nonrelativistic result already contains the skeleton of a relativistic cross section:

$$d\sigma ={} \frac{1}{\text{incident flux}} \times \left| \text{transition amplitude} \right|^2 \times d(\text{final phase space}).$$

In quantum field theory, the fixed-particle potential matrix element becomes an invariant amplitude $\mathcal M$, the density of states becomes Lorentz-invariant multiparticle phase space $d\Phi_n$, and particle production is a natural inelastic channel. Schematically,

$$d\sigma = \frac{1}{\mathcal F} \overline{\lvert\mathcal M\rvert^2} d\Phi_n.$$

The bar denotes the required spin sums and averages, and $\mathcal F$ is the relativistic flux factor. The logic survives; the normalization and kinematics change.

Exercises

1. Excitation threshold

A projectile with reduced mass $\mu$ scatters from a two-level target whose excited state lies an energy $\Delta$ above the ground state. The incoming relative wave number is $k_i$. Find the outgoing wave number for excitation, the threshold condition, and the kinematic factor multiplying $|f_{eg}|^2$.

Solution

Energy conservation gives

$$\frac{\hbar^2k_i^2}{2\mu} = \Delta + \frac{\hbar^2k_f^2}{2\mu}.$$

Therefore

$$k_f = \sqrt{ k_i^2 - \frac{2\mu\Delta}{\hbar^2} }.$$

The excitation channel is open only if

$$\frac{\hbar^2k_i^2}{2\mu} \ge \Delta.$$

For equal reduced masses, the differential cross section is

$$\frac{d\sigma_{e\leftarrow g}}{d\Omega} = \frac{k_f}{k_i} |f_{eg}|^2.$$

The kinematic factor vanishes at threshold, although a singular or resonant amplitude can modify the naive smooth turn-on.

2. Derive the velocity ratio

An outgoing channel has asymptotic radial wave

$$\psi_\beta = f_{\beta\alpha}(\Omega) \frac{e^{ik_\beta r}}{r}.$$

Use probability currents to derive the channel differential cross section.

Solution

At leading order in $1/r$, the outgoing radial current is

$$j_{\beta,r} = \frac{\hbar k_\beta}{\mu_\beta} \frac{|f_{\beta\alpha}|^2}{r^2} = v_\beta \frac{|f_{\beta\alpha}|^2}{r^2}.$$

The flux crossing area $r^2d\Omega$ per unit time is

$$dR_\beta = v_\beta |f_{\beta\alpha}|^2 d\Omega.$$

Dividing by the incident plane-wave current $j_\alpha=v_\alpha$ gives

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac{v_\beta}{v_\alpha} |f_{\beta\alpha}|^2.$$

3. Gaussian transition form factor

In the two-level model, let

$$W(r) = W_0e^{-r^2/(2a^2)}.$$

Find $\widetilde W(\mathbf q)$ and describe the angular dependence of the first Born excitation cross section.

Solution

The three-dimensional Gaussian transform factorizes into three one-dimensional integrals:

$$\widetilde W(\mathbf q) = (2\pi)^{3/2} a^3W_0 e^{-a^2q^2/2}.$$

Thus

$$\frac{d\sigma_{e\leftarrow g}^{(\mathrm B)}}{d\Omega} \propto{} \frac{k_e}{k_g} e^{-a^2q^2} \times \left| \langle e\rvert\mathcal O\lvert g\rangle \right|^2.$$

Because

$$q^2 = k_g^2+k_e^2-2k_gk_e\cos\theta,$$

large-angle scattering generally requires larger momentum transfer and is exponentially suppressed when $a$ is large compared with the relevant wavelength. Internal excitation also changes $k_e$, so the elastic identity $q=2k\sin(\theta/2)$ does not apply.

4. Coherent paths and orthogonal channels

Two dynamical paths lead to the same final channel with amplitudes $A_1$ and $A_2$. Two other paths lead to orthogonal final channels with amplitudes $B_1$ and $B_2$. What does an unresolved detector count in each case?

Solution

For the same final channel, amplitudes add before squaring:

$$\begin{aligned} P_A &= |A_1+A_2|^2\\ &= |A_1|^2+|A_2|^2 +2\operatorname{Re}(A_1A_2^*). \end{aligned}$$

For orthogonal final channels, tracing or summing over the unobserved channel label removes the cross term:

$$P_B = |B_1|^2+|B_2|^2.$$

Detector nonresolution does not make orthogonal records coherent. Interference requires indistinguishable alternatives ending in the same quantum state, or in final states with nonzero overlap.

5. Threshold exponents

Near an endothermic threshold, suppose the entrance momentum is finite and the short-range transition amplitude is smooth. Determine the energy dependence for exit $s$ and $p$ waves.

Solution

Let

$$\varepsilon = E-E_\beta^{\mathrm{th}}>0.$$

Then $k_\beta\propto\varepsilon^{1/2}$. The Wigner law gives

$$\sigma^{(\ell_\beta)} \propto k_\beta^{2\ell_\beta+1}.$$

For $\ell_\beta=0$,

$$\sigma^{(s)} \propto \varepsilon^{1/2}.$$

For $\ell_\beta=1$,

$$\sigma^{(p)} \propto \varepsilon^{3/2}.$$

Long-range interactions or nearby poles can change these powers or dominate the observed line shape.

6. Elastic deficit in one partial wave

For one incoming channel $\alpha$, suppose the full open-channel partial-wave $S$-matrix is unitary and

$$S_{\alpha\alpha}^{(\ell)} = \eta_\ell e^{2i\delta_\ell}.$$

Show that the summed transition probability into all other open channels is $1-\eta_\ell^2$, and find the corresponding inelastic partial cross section in the standard flux-normalized convention.

Solution

Unitarity of the incoming column gives

$$\sum_\beta \left| S_{\beta\alpha}^{(\ell)} \right|^2 =1.$$

Separating the elastic term,

$$\sum_{\beta\ne\alpha} \left| S_{\beta\alpha}^{(\ell)} \right|^2 = 1- \left| S_{\alpha\alpha}^{(\ell)} \right|^2 = 1-\eta_\ell^2.$$

The summed inelastic partial cross section is therefore

$$\sigma_{\mathrm{inel}}^{(\ell)} = \frac{\pi}{k_\alpha^2} (2\ell+1) (1-\eta_\ell^2).$$

The elastic element can have magnitude below one while the complete open-channel $S$-matrix remains unitary.

References

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• C. J. Joachain, Quantum Collision Theory, North-Holland, 1975.
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• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed., Cambridge University Press, 2012.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Wiley, 1972.
• E. P. Wigner, “On the Behavior of Cross Sections Near Thresholds,” Physical Review 73, 1002–1009 (1948), doi:10.1103/PhysRev.73.1002.