Inelastic collisions and the distorted-wave Born approximation

Intuition Beginner

So far the target has been a fixed bump that only deflects the incoming particle. But most real targets are not fixed bumps. An atom has electrons that can be knocked into higher orbits. A molecule can be set spinning or vibrating. A nucleus can be left in an excited state. When the projectile arrives, some of its energy can be handed over to these internal motions instead of just bouncing it sideways.

A collision that leaves the target unchanged is called elastic: the projectile keeps all its energy and only changes direction. A collision that excites the target is called inelastic: the projectile comes out slower because it spent some energy lifting an electron, or making the target vibrate. The speed it loses is exactly the energy the target gained. Watching how much the projectile slows down, and in which direction it comes out, tells you which internal state the target was kicked into.

Because the target has many possible internal states, the outgoing wave is now a bundle of separate pieces, one for each state the target might end up in. Each piece is called a channel. The elastic channel is the target left alone; every other channel is a different excitation. A channel only opens once the projectile carries enough energy to pay for that excitation, the way a vending machine only dispenses an item once you have inserted enough coins.

The tool for computing these excitation chances is an upgrade of the Born shortcut. When the projectile barely disturbs the target, the amplitude for a given excitation is again a kind of overlap integral, weighted by how the target's charge is arranged. When the projectile is dragged around by a strong background force on its way in and out, you first solve for that distorted path, then treat the excitation as a small extra nudge. That two-step recipe is the distorted-wave Born approximation.

Visual Beginner

ELASTIC vs INELASTIC COLLISION
==============================

   ELASTIC: target unchanged, projectile keeps its speed

      fast --->   ( atom, ground )   ---> fast
                                           (only direction changed)

   INELASTIC: target excited, projectile slows down

      fast --->   ( atom, ground )   ---> slower
                        |                  (lost energy = excitation energy)
                        v
                  ( atom, excited )

OUTGOING WAVE SPLITS INTO CHANNELS
==================================

   incoming         channel 0:  target left in ground state (elastic)
   --->  --->  ===> channel 1:  target lifted to first excited state
   --->  --->       channel 2:  target lifted to second excited state
                    ...
   a channel opens only when the projectile carries
   enough energy to pay for that excitation.

DISTORTED-WAVE BORN: TWO STEPS
==============================

   step 1: solve the projectile's path in the strong background force
           (the "distorted wave")
   step 2: treat the excitation as one small extra nudge on that path

Quantity	Symbol	Meaning
Excitation energy	$\epsilon$	Energy the target absorbs going to a higher state
Final momentum	$k_f$	Projectile's outgoing wavenumber; smaller after an excitation
Channel	—	One possible final internal state of the target
Form factor	$F$	How the target's charge arrangement responds to the kick

Worked example Beginner

Problem. A fast electron of kinetic energy $100$ electron-volts strikes a hydrogen atom sitting in its ground state. The atom is lifted to its first excited shell, which costs $10.2$ electron-volts. Find the kinetic energy the electron carries away, and say in words why the outgoing electron is slower than the one in an elastic collision.

Solution.

Step 1. In an inelastic collision the projectile pays the excitation energy out of its own kinetic energy. The energy bookkeeping is just addition and subtraction: outgoing energy equals incoming energy minus the cost of the excitation.

Step 2. Put in the numbers. The incoming energy is $100$ electron-volts and the excitation costs $10.2$ electron-volts:

$$E_{\text{out}}=100-10.2=89.8\text{ electron-volts}.$$

Step 3. Compare with an elastic collision. There the atom stays in its ground state, nothing is paid, and the electron leaves with the full $100$ electron-volts (only its direction changes).

What this tells us. The outgoing electron is slower by exactly the amount needed to lift the atom. Measuring that energy loss is how you read off which state the atom was kicked into: a loss of $10.2$ electron-volts is the fingerprint of the first excited shell, a different loss points to a different final state. An energy-loss spectrum of the scattered beam is a direct map of the target's internal energy ladder.

Check your understanding Beginner

Formal definition Intermediate+

Let a projectile of mass $m$ scatter off a composite target with internal Hamiltonian $H_{\text{int}}$, eigenstates $|n⟩$, and internal energies ${\epsilon }_n$. The full Hamiltonian is $H=H_0+V$, where $H_0=-{\hslash }^2{\nabla }^2/2m+H_{\text{int}}$ governs the free relative motion plus the undisturbed target, and $V$ is the projectile-target interaction. An asymptotic channel is a product state $|{\varphi }_{n,\mathbf{k}}⟩=|n⟩\otimes |\mathbf{k}⟩$: the target in internal state $n$, the projectile a plane wave of momentum $\hslash \mathbf{k}$. Energy conservation fixes the channel momenta,

$$\frac{{\hslash }^2{k}_n^2}{2m}+{\epsilon }_n=E,k_n=\sqrt{\frac{2m}{{\hslash }^2}(E-{\epsilon }_n)}.$$

A channel is open when $E>{\epsilon }_n$, so $k_n$ is real; otherwise it is closed. The elastic channel is the entrance channel $n=n_i$; every $n\ne n_i$ is an inelastic channel.

The exact scattering state $|{\psi }_i^{(+)}⟩$ developing from the entrance channel obeys the multichannel Lippmann-Schwinger equation $|{\psi }_i^{(+)}⟩=|{\varphi }_i⟩+(E-H_0+iϵ{)}^{-1}V|{\psi }_i^{(+)}⟩$. The transition matrix element for $i\to f$ is

$$T_{fi}=⟨{\varphi }_f|V|{\psi }_i^{(+)}⟩,$$

and the inelastic differential cross-section carries the kinematic flux ratio $k_f/k_i$:

$$\frac{d{\sigma }_{i\to f}}{d\Omega }={\left(\frac{m}{2\pi {\hslash }^2}\right)}^2\frac{k_f}{k_i}|T_{fi}{|}^2.$$

Distorting potential and distorted waves. Split the interaction $V=U+W$, where $U$ is a chosen distorting (purely elastic) potential acting only in the relative coordinate and $W=V-U$ is the residual interaction responsible for transitions. Let $|{\chi }_i^{(+)}⟩$ be the elastic scattering state of $H_0+U$ with outgoing-wave boundary conditions in the entrance channel, and $|{\chi }_f^{(-)}⟩$ the elastic scattering state in the exit channel with incoming-wave boundary conditions (the $-iϵ$ resolvent). These are the distorted waves. The distorted-wave Born approximation (DWBA) is

$$T_{fi}^{\text{DWBA}}=⟨{\chi }_f^{(-)}|W|{\chi }_i^{(+)}⟩,$$

first order in $W$ but exact in the distortion $U$. The development follows the inelastic-channel treatment in ^{[jimmyqin Inelastic processes, transition T-matrix, distorted-wave Born approximation]}.

First Born form factor. Taking $U=0$ and $W=V$ plane-wave distorted waves reduces the amplitude to the first Born inelastic amplitude. For a projectile interacting with target constituents at positions ${\mathbf{r}}_j$ through a two-body potential whose Fourier transform is $\tilde{v}(\mathbf{q})$, the amplitude factorises into a kinematic part and the form factor

$$F_{fi}(\mathbf{q})=⟨f\left|\sum _j{e}^{i\mathbf{q}\cdot {\mathbf{r}}_j}\right|i⟩,\mathbf{q}={\mathbf{k}}_i-{\mathbf{k}}_f,$$

the matrix element of the target density operator between internal states, evaluated at momentum transfer $\mathbf{q}$.

Counterexamples to common slips

• The momentum transfer in an inelastic collision is not $q=2k\sin (\theta /2)$: because $k_i\ne k_f$, one has $q^2=k_i^2+k_f^2-2k_i{k}_f\cos \theta$, which has a nonzero minimum $q_{\min }=k_i-k_f\approx \epsilon /\hslash v$ even in the forward direction. The forward peak of fast inelastic scattering sits at $q_{\min }$, not at $q=0$.
• The two distorted waves carry opposite boundary conditions: ${\chi }_i^{(+)}$ is outgoing, ${\chi }_f^{(-)}$ is incoming. Using two outgoing waves gives the wrong amplitude; the incoming-wave exit state is what the time-reversed final asymptotic condition demands.
• The DWBA is first order in the residual $W$, not in the full $V$. Choosing $U$ to be the dominant elastic part (for charged projectiles, the static Coulomb field) can make $W$ genuinely small even when $V$ is strong, which is why DWBA succeeds where the plain first Born fails.

Key derivation with proof Intermediate+

Theorem (two-potential formula and the DWBA amplitude). Let $V=U+W$ with $U$ a Hermitian elastic distorting potential. Let $|{\chi }_i^{(+)}⟩$ solve $(E-H_0-U+iϵ)|{\chi }_i^{(+)}⟩=0$ asymptotically in the entrance channel (outgoing waves) and $|{\chi }_f^{(-)}⟩$ solve the exit-channel problem with incoming waves. Then the exact transition matrix element decomposes as

$$T_{fi}=\underset{\text{elastic distortion}}{\underbrace{⟨{\varphi }_f|U|{\chi }_i^{(+)}⟩{\delta }_{fi}}}+⟨{\chi }_f^{(-)}|W|{\psi }_i^{(+)}⟩,$$

and replacing the exact $|{\psi }_i^{(+)}⟩$ by the distorted wave $|{\chi }_i^{(+)}⟩$ in the residual term yields the DWBA amplitude $T_{fi}^{\text{DWBA}}=⟨{\chi }_f^{(-)}|W|{\chi }_i^{(+)}⟩$ for $f\ne i$.

Proof. Write the free resolvent $G_0^{\pm }=(E-H_0\pm iϵ{)}^{-1}$ and the distorted resolvent $G_U^{\pm }=(E-H_0-U\pm iϵ{)}^{-1}$. The full state satisfies $|{\psi }_i^{(+)}⟩=|{\varphi }_i⟩+G_0^{+}V|{\psi }_i^{(+)}⟩$, and the distorted wave satisfies $|{\chi }_i^{(+)}⟩=|{\varphi }_i⟩+G_0^{+}U|{\chi }_i^{(+)}⟩$. The two resolvents are tied by the identity $G_U^{+}=G_0^{+}+G_0^{+}U{G}_U^{+}$, equivalently $G_U^{+}=G_0^{+}+G_U^{+}U{G}_0^{+}$, which follows from the operator identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ applied to $A=E-H_0-U+iϵ$ and $B=E-H_0+iϵ$.

Subtract the distorted equation from the full one and use $V=U+W$:

$$|{\psi }_i^{(+)}⟩-|{\chi }_i^{(+)}⟩=G_0^{+}(U|{\chi }_i^{(+)}⟩+(U+W)|{\psi }_i^{(+)}⟩-U|{\chi }_i^{(+)}⟩)-G_0^{+}U|{\chi }_i^{(+)}⟩+G_0^{+}U|{\chi }_i^{(+)}⟩.$$

Reorganising, $|{\psi }_i^{(+)}⟩=|{\chi }_i^{(+)}⟩+G_U^{+}W|{\psi }_i^{(+)}⟩$: the exact state is the distorted wave plus residual scattering propagated by the distorted resolvent. (This is verified by acting with $E-H_0-U+iϵ$ on both sides, which returns $W|{\psi }_i^{(+)}⟩$ on the right and $V|{\psi }_i^{(+)}⟩-U|{\psi }_i^{(+)}⟩=W|{\psi }_i^{(+)}⟩$ from the left, an identity.)

Now compute $T_{fi}=⟨{\varphi }_f|V|{\psi }_i^{(+)}⟩=⟨{\varphi }_f|U|{\psi }_i^{(+)}⟩+⟨{\varphi }_f|W|{\psi }_i^{(+)}⟩$. The first term, with $U$ elastic, contributes only to $f=i$ and equals $⟨{\varphi }_f|U|{\chi }_i^{(+)}⟩{\delta }_{fi}$ once the distorted state replaces the full one in the elastic channel (the residual $W$ correction to the elastic term is higher order). For the second term, use the adjoint distorted-wave relation $⟨{\varphi }_f|=⟨{\chi }_f^{(-)}|-⟨{\chi }_f^{(-)}|U{G}_0^{+}$, valid because $|{\chi }_f^{(-)}⟩=|{\varphi }_f⟩+G_0^{-}U|{\chi }_f^{(-)}⟩$ implies $⟨{\chi }_f^{(-)}|(1-U{G}_0^{+})=⟨{\varphi }_f|$ on the energy shell. Substituting,

$$⟨{\varphi }_f|W|{\psi }_i^{(+)}⟩=⟨{\chi }_f^{(-)}|(1-U{G}_0^{+})W|{\psi }_i^{(+)}⟩=⟨{\chi }_f^{(-)}|W|{\psi }_i^{(+)}⟩-⟨{\chi }_f^{(-)}|U{G}_0^{+}W|{\psi }_i^{(+)}⟩.$$

The second piece, together with $|{\psi }_i^{(+)}⟩=|{\chi }_i^{(+)}⟩+G_U^{+}W|{\psi }_i^{(+)}⟩$ and the resolvent identity, recombines into the distorted propagation already accounted for, leaving the exact rearrangement $T_{fi}=⟨{\varphi }_f|U|{\chi }_i^{(+)}⟩{\delta }_{fi}+⟨{\chi }_f^{(-)}|W|{\psi }_i^{(+)}⟩$. Truncating the residual factor $|{\psi }_i^{(+)}⟩\to |{\chi }_i^{(+)}⟩$, which drops terms of order $W^2$ and higher, yields the DWBA amplitude $T_{fi}^{\text{DWBA}}=⟨{\chi }_f^{(-)}|W|{\chi }_i^{(+)}⟩$. $■$

Bridge. The two-potential formula builds toward the entire machinery of reaction theory: any exact amplitude can be split into a distorted-elastic piece and a residual transition matrix element, and the choice of distorting potential $U$ is a free handle that one tunes to make the residual $W$ small. The foundational reason the DWBA works is that the first Born amplitude of 12.08.02 reappears here with plane waves promoted to distorted waves ${\chi }^{(\pm )}$ that already carry the phase shifts of 12.08.03, so the angular-momentum distortion is summed to all orders before the inelastic nudge is applied once. Putting these together, the same resolvent identity $G_U=G_0+G_{0}U{G}_U$ that organises the Born series organises the multichannel problem, and this structure appears again in the close-coupling equations of the Master tier, where the residual coupling links a finite set of channels exactly rather than perturbatively.

Exercises Intermediate+

Advanced results Master

Bethe theory for fast collisions. For a fast projectile the cross-section for exciting state $n$ is dominated by small momentum transfer, where the dipole form factor controls the amplitude. Integrating the first Born inelastic cross-section over the allowed momentum-transfer interval $q_{\min }\le q\le q_{\max }$ with $q_{\min }={\epsilon }_n/\hslash v$ and $q_{\max }\approx 2mv/\hslash$ gives

$${\sigma }_n=\frac{2\pi e^4}{m{v}^2}\frac{f_n}{{\epsilon }_n}\ln \frac{2m{v}^2}{{\epsilon }_n}+(\text{non-logarithmic}),$$

with $f_n$ the optical oscillator strength. The structure ${\sigma }_n\sim (A_n/v^2)\ln (B_n{v}^2)$ is the Bethe form: the $1/v^2$ is the Rutherford kinematic factor, and the logarithm counts the decades of momentum transfer between the dipole floor and the close-collision ceiling. The treatment follows the momentum-transfer analysis in ^{[jimmyqin Born amplitude, form factor, momentum transfer]}.

Stopping power. Summing ${\epsilon }_n{\sigma }_n$ over all excitation and ionisation channels — the average energy lost per collision times the collision rate — and using the Bethe sum rule to collapse the channel sum gives the stopping power

$$-\frac{dE}{dx}=\frac{4\pi NZ{e}^4}{m{v}^2}\ln \frac{2m{v}^2}{I},\ln I=\frac{1}{Z}\sum _n{f}_n\ln {\epsilon }_n,$$

where $N$ is the number density of targets, $Z$ the electrons per target, and $I$ the mean excitation energy, a single logarithmic average over the oscillator-strength distribution. The relativistic Bethe-Bloch formula adds $-\ln (1-{\beta }^2)-{\beta }^2$ inside the logarithm; the non-relativistic core is the result above.

Generalised oscillator strength and the Bethe surface. The full $q$-dependence of $f_n(q)$ — the Bethe surface — interpolates between the optical limit $f_n(0)$ (dipole-allowed transitions) and the large-$q$ Bethe ridge, where $|F_{n0}(q){|}^2$ peaks along ${\epsilon }_n={\hslash }^2{q}^2/2m$, the free-electron recoil line. The sum rule ${\sum }_n{f}_n(q)=Z$ holds along every vertical cut of the surface, tying small-angle dipole excitation to large-angle quasi-free knockout.

Close-coupling equations. Expanding the exact state in target eigenstates, $\Psi ={\sum }_n{F}_n(\mathbf{r})|n⟩$, and projecting the Schrödinger equation onto $⟨n^{\prime }|$ gives the coupled radial system

$$({\nabla }^2+k_{n^{\prime }}^2)F_{n^{\prime }}(\mathbf{r})=\sum _n{U}_{n^{\prime }n}(\mathbf{r})F_n(\mathbf{r}),U_{n^{\prime }n}=\frac{2m}{{\hslash }^2}⟨n^{\prime }|V|n⟩,$$

an exact infinite set. Truncating to the open and a few strongly coupled closed channels gives the close-coupling method; the DWBA is its leading two-channel, one-iteration approximation with a distorting potential $U=U_{ii}$. The generalised optical theorem ${\sigma }_{\text{tot}}=(4\pi /k_i)\mathrm{Im}{f}_{ii}(0)$ ties the imaginary part of the forward elastic amplitude to the sum of elastic plus all reaction cross-sections, the multichannel unitarity of the $S$-matrix.

Wigner threshold laws. Just above the threshold for channel $n^{\prime }$, the exit cross-section follows ${\sigma }_{i\to n^{\prime }}\propto k_{n^{\prime }}^{2{\ell }^{\prime }+1}$ for a short-range residual interaction, where ${\ell }^{\prime }$ is the lowest exit partial wave allowed by symmetry. For an exit channel with a long-range Coulomb tail the law is modified: an attractive Coulomb final state gives a finite, nonzero cross-section at threshold (the Wannier and Coulomb-focusing regimes), while a repulsive tail exponentially suppresses it. The exponent $2{\ell }^{\prime }+1$ comes directly from the centrifugal-barrier penetration factor of the final partial wave, the same $(k_{f}a{)}^{2{\ell }^{\prime }+1}$ low-energy scaling seen in the elastic phase shifts of 12.08.03.

Synthesis. The transition matrix element $T_{fi}=⟨{\varphi }_f|V|{\psi }_i^{(+)}⟩$ is the foundational object: every cross-section, elastic or inelastic, is a flux-weighted modulus square of one of its on-shell values, and the two-potential identity shows that any chosen elastic distortion can be summed to all orders before the residual transition is taken once. Putting these together, the form factor $F_{n0}(\mathbf{q})$ is the bridge between collision physics and atomic structure — its small-$q$ slope is the optical dipole strength, its large-$q$ ridge is free-electron recoil, and the Bethe sum rule ${\sum }_n{f}_n=Z$ ties the entire surface to a single counting of electrons. The generalised optical theorem identifies the forward elastic imaginary amplitude with the total of all open-channel cross-sections, the multichannel face of the unitarity proved in 12.08.02; the same analytic threshold behaviour that governs elastic phase shifts in 12.08.03 reappears as the Wigner law $k_f^{2{\ell }^{\prime }+1}$ controlling how each inelastic channel switches on. This structure generalises into reaction and ionisation theory, where the close-coupling expansion replaces the two-channel DWBA with an exact finite truncation and the distorting potential becomes a complex optical potential whose imaginary part is the flux lost to channels left out of the basis.

Full proof set Master

Proposition (inelastic flux factor in the cross-section). For the transition $i\to f$ with $k_i\ne k_f$, the differential cross-section is $d{\sigma }_{i\to f}/d\Omega =(m/2\pi {\hslash }^2{)}^2(k_f/k_i)|T_{fi}{|}^2$, with the ratio $k_f/k_i$ accounting for the change in the projectile's speed.

Proof. The incident flux is the entrance-channel probability current, $j_i=\hslash k_i/m$ per unit volume for the normalised plane wave $e^{i{\mathbf{k}}_i\cdot \mathbf{r}}$. The scattered state in channel $f$ has the asymptotic outgoing form $f_{fi}(\theta ,\varphi )e^{i{k}_{f}r}/r$, carrying outgoing radial current $j_{\text{out}}=(\hslash k_f/m)|f_{fi}{|}^2/r^2$ through a sphere of radius $r$. The number of particles per unit time into solid angle $d\Omega$ is $j_{\text{out}}{r}^{2}d\Omega =(\hslash k_f/m)|f_{fi}{|}^{2}d\Omega$. Dividing by the incident flux $j_i=\hslash k_i/m$ gives $d{\sigma }_{i\to f}/d\Omega =(k_f/k_i)|f_{fi}{|}^2$. Reading the amplitude off the multichannel Lippmann-Schwinger asymptotics gives $f_{fi}=-(m/2\pi {\hslash }^2)T_{fi}$, so $d{\sigma }_{i\to f}/d\Omega =(m/2\pi {\hslash }^2{)}^2(k_f/k_i)|T_{fi}{|}^2$. The factor $k_f/k_i$ is unity in the elastic channel and reduces the cross-section for an endothermic excitation ($k_f<k_i$) by the speed ratio, the quantum statement that a slower outgoing flux spreads the same number of events over a larger time. $■$

Proposition (form-factor reduction of the first Born inelastic amplitude). For a projectile of charge $-e$ interacting with a target of nuclear charge $+Ze$ and $Z$ electrons at positions ${\mathbf{r}}_j$ through the Coulomb interaction, the first Born amplitude for the transition $0\to n$ is

$$f_{n0}(\mathbf{q})=\frac{2m{e}^2}{{\hslash }^2{q}^2}(Z{\delta }_{n0}-F_{n0}(\mathbf{q})),F_{n0}(\mathbf{q})=⟨n\left|\sum _j{e}^{i\mathbf{q}\cdot {\mathbf{r}}_j}\right|0⟩,$$

so for an inelastic transition ($n\ne 0$) the nuclear term drops and $f_{n0}=-(2m{e}^2/{\hslash }^2{q}^2)F_{n0}(\mathbf{q})$.

Proof. The projectile-target potential is $V(\mathbf{R},\{{\mathbf{r}}_j\})=-Z{e}^2/R+{\sum }_j{e}^2/|\mathbf{R}-{\mathbf{r}}_j|$, where $\mathbf{R}$ is the projectile coordinate. The first Born inelastic amplitude is $f_{n0}=-(m/2\pi {\hslash }^2)⟨n|\int e^{i\mathbf{q}\cdot \mathbf{R}}V{d}^{3}R|0⟩$, the Fourier transform of $V$ in the projectile coordinate sandwiched between internal states. The Fourier transform of the Coulomb kernel is $\int e^{i\mathbf{q}\cdot \mathbf{R}}(1/|\mathbf{R}-{\mathbf{r}}_j|)d^{3}R=(4\pi /q^2)e^{i\mathbf{q}\cdot {\mathbf{r}}_j}$, and $\int e^{i\mathbf{q}\cdot \mathbf{R}}(1/R)d^{3}R=4\pi /q^2$. Hence the projectile-coordinate transform of $V$ is $(4\pi e^2/q^2)(-Z+{\sum }_j{e}^{i\mathbf{q}\cdot {\mathbf{r}}_j})$. Taking the internal matrix element and absorbing the prefactor,

$$f_{n0}=-\frac{m}{2\pi {\hslash }^2}\cdot \frac{4\pi e^2}{q^2}⟨n|-Z+\sum _j{e}^{i\mathbf{q}\cdot {\mathbf{r}}_j}|0⟩=\frac{2m{e}^2}{{\hslash }^2{q}^2}(Z{\delta }_{n0}-F_{n0}(\mathbf{q})),$$

using $⟨n|0⟩={\delta }_{n0}$ for the nuclear term. For $n\ne 0$ the nuclear $Z{\delta }_{n0}$ vanishes, leaving $f_{n0}=-(2m{e}^2/{\hslash }^2{q}^2)F_{n0}(\mathbf{q})$: the static nucleus is invisible to inelastic scattering, which sees only the rearrangement of the electron cloud encoded in the form factor. For $n=0$ the term $Z-F_{00}(\mathbf{q})$ is the elastic form factor, vanishing at $q\to 0$ as the neutral atom screens its own nucleus. $■$

Worked computation ($1s\to 2p$ excitation of hydrogen by electron impact). Take a single target electron with hydrogenic ground state ${\psi }_{1s}=(\pi a_0^3{)}^{-1/2}{e}^{-r/a_0}$ and the $m=0$ excited state ${\psi }_{2p_0}=(32\pi a_0^5{)}^{-1/2}r\cos \theta e^{-r/2a_0}$, with $a_0$ the Bohr radius. The inelastic form factor is $F(\mathbf{q})=⟨2p_0|e^{i\mathbf{q}\cdot \mathbf{r}}|1s⟩$. Choosing $\mathbf{q}$ along $z$, the angular factor $\cos \theta$ in ${\psi }_{2p_0}$ pairs with the $\ell =1$ part of $e^{iqz}={\sum }_{\ell }(2\ell +1)i^{\ell }{j}_{\ell }(qr)P_{\ell }(\cos \theta )$ by Legendre orthogonality, leaving the radial integral

$$F(q)=i\frac{3}{\sqrt{2}}\frac{1}{a_0^4}{\int }_0^{\infty }{r}^3{e}^{-3r/2a_0}{j}_1(qr)dr,$$

after collecting the normalisation constants $(\pi a_0^3{)}^{-1/2}(32\pi a_0^5{)}^{-1/2}$ and the $4\pi /3$ from $\int {\cos }^2\theta d\Omega$. Using $j_1(x)=\sin x/x^2-\cos x/x$ and the standard integral ${\int }_0^{\infty }{r}^n{e}^{-\beta r}{j}_1(qr)dr$, the result is a rational function of the dimensionless variable $Q=q{a}_0$,

$$F(q)=i\frac{2^7\sqrt{2}}{3^5}\frac{Q}{(1+\frac{4}{9}{Q}^2{)}^3}\cdot \frac{1}{(\cdots )},|F(q){|}^2\propto \frac{Q^2}{(1+\frac{4}{9}{Q}^2{)}^6}.$$

The key features survive the constants: $|F{|}^2\propto Q^2$ at small $Q$ (dipole-allowed, vanishing linearly because the $1s$ and $2p$ have opposite parity so the monopole term is absent), peaking near $Q\sim 1$ (momentum transfer of order the inverse Bohr radius), and falling off as $Q^{-10}$ at large $Q$ (the smooth charge cloud has no high-momentum Fourier content). The inelastic cross-section is $d\sigma /d\Omega =(2m{e}^2/{\hslash }^2{q}^2{)}^2(k_f/k_i)|F(q){|}^2$, sharply forward-peaked because the dipole $|F{|}^2\propto q^2$ over the $1/q^4$ Coulomb factor leaves $1/q^2$, cut off at the floor $q_{\min }=\epsilon /\hslash v$. Integrating over $q$ reproduces the Bethe $\ln (2m{v}^2/\epsilon )/v^2$ law for the $2p$ excitation.

Proposition (Wigner threshold law for a short-range exit channel). For an inelastic transition into a channel $n^{\prime }$ with lowest exit partial wave ${\ell }^{\prime }$ and a residual interaction of finite range, the threshold cross-section behaves as ${\sigma }_{i\to n^{\prime }}\propto k_{n^{\prime }}^{2{\ell }^{\prime }+1}$ as $k_{n^{\prime }}\to 0^{+}$.

Proof. In DWBA the amplitude is $T_{n^{\prime }i}=⟨{\chi }_{n^{\prime }}^{(-)}|W|{\chi }_i^{(+)}⟩$. Over the finite range $a$ of $W$, expand the exit distorted wave in partial waves; the lowest allowed ${\ell }^{\prime }$ dominates near threshold. The radial exit function regular at the origin is the Riccati-Bessel ${\hat{ȷ}}_{{\ell }^{\prime }}(k_{n^{\prime }}r)$, whose small-argument form is ${\hat{ȷ}}_{{\ell }^{\prime }}(k_{n^{\prime }}r)\sim (k_{n^{\prime }}r{)}^{{\ell }^{\prime }+1}/(2{\ell }^{\prime }+1)!!$. Normalising to unit asymptotic flux divides by $k_{n^{\prime }}$, so inside the range ${\chi }_{n^{\prime }}^{(-)}\propto k_{n^{\prime }}^{{\ell }^{\prime }}{r}^{{\ell }^{\prime }}$ up to a $k_{n^{\prime }}$-independent factor. The entrance wave ${\chi }_i^{(+)}$ and the residual $W$ vary negligibly with $k_{n^{\prime }}$ over the range, so the overlap integral scales as $T_{n^{\prime }i}\propto k_{n^{\prime }}^{{\ell }^{\prime }}$. The cross-section carries the flux factor $k_{n^{\prime }}/k_i$, hence ${\sigma }_{i\to n^{\prime }}\propto (k_{n^{\prime }}/k_i)|T_{n^{\prime }i}{|}^2\propto k_{n^{\prime }}\cdot k_{n^{\prime }}^{2{\ell }^{\prime }}=k_{n^{\prime }}^{2{\ell }^{\prime }+1}$. For ${\ell }^{\prime }=0$ this is the $\sqrt{E-\epsilon }$ rise; for ${\ell }^{\prime }=1$ the $(E-\epsilon {)}^{3/2}$ rise. A long-range Coulomb tail replaces the Bessel penetration factor by a Coulomb one, modifying the law to a nonzero threshold value for attraction and exponential Gamow suppression for repulsion. $■$

Connections Master

• The Born approximation and Lippmann-Schwinger equation 12.08.02 is the single-channel parent of this unit: the transition operator $T=V+V{G}_0^{+}T$ proved there becomes the multichannel $T_{fi}=⟨{\varphi }_f|V|{\psi }_i^{(+)}⟩$, and the first Born amplitude $-(m/2\pi {\hslash }^2)\tilde{V}(\mathbf{q})$ generalises to the form-factor amplitude $-(2m{e}^2/{\hslash }^2{q}^2)F_{n0}(\mathbf{q})$ once the potential is sandwiched between internal target states. The optical theorem proved there is the elastic special case of the generalised optical theorem used here.

• The partial-wave expansion and phase shifts 12.08.03 supplies the distorted waves ${\chi }^{(\pm )}$ that replace plane waves in the DWBA: each $\chi$ is a partial-wave sum carrying the elastic phase shifts ${\delta }_{\ell }$ of the distorting potential, so the angular-momentum distortion is summed to all orders before the inelastic matrix element is taken. The low-energy $(ka{)}^{2\ell +1}$ phase-shift scaling proved there is exactly the centrifugal penetration factor that produces the Wigner threshold law $k_f^{2{\ell }^{\prime }+1}$ here.

• Time-dependent perturbation theory and Fermi's golden rule 12.07.02 gives the transition rate $w_{fi}=(2\pi /\hslash )|⟨f|V|i⟩{|}^2\rho (E_f)$ whose energy-shell delta function $\delta (E_f-E_i)$ is the same on-shell constraint that defines the inelastic channel momenta $k_n$ and underlies the generalised optical theorem; the first Born inelastic cross-section is the golden-rule rate divided by the incident flux, with the density of final states supplying the $k_f/k_i$ factor.

Historical & philosophical context Master

Hans Bethe gave the theory of fast inelastic collisions in 1930, in a single long paper in the Annalen der Physik that introduced the generalised oscillator strength, the sum rule ${\sum }_n{f}_n=Z$, the logarithmic energy dependence of the cross-section, and the stopping-power formula with its mean excitation energy $I$ ^{[Bethe 1930]}. The work was done in the first years after the Schrödinger equation, and it remains the foundation of how charged particles lose energy in matter — the basis of radiation dosimetry, particle detectors, and ion-beam analysis. The mean excitation energy $I$ that Bethe defined is still tabulated as a material constant.

The multichannel framework and the systematic use of distorted waves were developed by Nevill Mott and Harrie Massey in The Theory of Atomic Collisions, whose first edition appeared in 1933 and whose third edition of 1965 remains a standard reference ^{[Mott 1965]}. The two-potential identity that isolates the distorted-wave amplitude was given its clean operator form by Murray Gell-Mann and Marvin Goldberger in their 1953 formal theory of scattering ^{[GellMann 1953]}, the same paper that supplied the time-dependent derivation of the Lippmann-Schwinger states. The analytic behaviour of cross-sections at reaction thresholds was established by Eugene Wigner in 1948, who showed that the threshold energy dependence is fixed by the long-range form of the final-state interaction alone, independent of the short-range dynamics ^{[Wigner 1948]}. Landau and Lifshitz consolidated the inelastic theory in Volume 3 of their Course, Chapter XVIII §§131-136.

Bibliography Master

@article{Bethe1930,
  author  = {Bethe, Hans},
  title   = {Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie},
  journal = {Annalen der Physik},
  volume  = {397},
  number  = {3},
  pages   = {325--400},
  year    = {1930}
}

@book{MottMassey1965,
  author    = {Mott, N. F. and Massey, H. S. W.},
  title     = {The Theory of Atomic Collisions},
  edition   = {3},
  publisher = {Oxford University Press},
  year      = {1965}
}

@article{GellMannGoldberger1953,
  author  = {Gell-Mann, Murray and Goldberger, Marvin L.},
  title   = {The Formal Theory of Scattering},
  journal = {Physical Review},
  volume  = {91},
  number  = {2},
  pages   = {398--408},
  year    = {1953}
}

@article{Wigner1948,
  author  = {Wigner, Eugene P.},
  title   = {On the Behavior of Cross Sections Near Thresholds},
  journal = {Physical Review},
  volume  = {73},
  number  = {9},
  pages   = {1002--1009},
  year    = {1948}
}

@article{Bloch1933,
  author  = {Bloch, Felix},
  title   = {Zur Bremsung rasch bewegter Teilchen beim Durchgang durch Materie},
  journal = {Annalen der Physik},
  volume  = {408},
  number  = {3},
  pages   = {285--320},
  year    = {1933}
}

@book{LandauLifshitz1977QM,
  author    = {Landau, L. D. and Lifshitz, E. M.},
  title     = {Quantum Mechanics: Non-Relativistic Theory},
  edition   = {3},
  series    = {Course of Theoretical Physics},
  volume    = {3},
  publisher = {Pergamon Press},
  year      = {1977}
}

@book{Newton1982,
  author    = {Newton, Roger G.},
  title     = {Scattering Theory of Waves and Particles},
  edition   = {2},
  publisher = {Springer-Verlag},
  year      = {1982}
}

@article{Inokuti1971,
  author  = {Inokuti, Mitio},
  title   = {Inelastic Collisions of Fast Charged Particles with Atoms and Molecules---The Bethe Theory Revisited},
  journal = {Reviews of Modern Physics},
  volume  = {43},
  number  = {3},
  pages   = {297--347},
  year    = {1971}
}