Scattering Cross Section

Purpose

A cross section converts outgoing event rate into an effective area by dividing by incident flux. In the standard nonrelativistic elastic convention,

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r}, \qquad r\to\infty,$$

and

$$\frac{d\sigma}{d\Omega} = \lvert f(\theta,\phi)\rvert^2.$$

This compact formula assumes the incident and outgoing waves belong to the same channel and therefore carry the same speed. For an inelastic channel $\beta\leftarrow\alpha$,

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac{v_\beta}{v_\alpha} \lvert f_{\beta\alpha}(\theta,\phi)\rvert^2.$$

The canonical derivation from flux is at Differential and Total Cross Sections. This card keeps the normalization, channel sums, partial-wave forms, and experimental counting factors together.

At a glance

Observable	Formula
Elastic differential cross section	$d\sigma/d\Omega=\lvert f\rvert^2$
Channel differential cross section	$d\sigma_{\beta\leftarrow\alpha}/d\Omega=(v_\beta/v_\alpha)\lvert f_{\beta\alpha}\rvert^2$
Acceptance-integrated cross section	$\sigma_{\mathcal A}=\int_{\mathcal A}d\Omega\,\epsilon(\Omega)\,d\sigma/d\Omega$
Total elastic cross section	$\sigma_{\mathrm{el}}=\int d\Omega\,\lvert f_{\alpha\alpha}\rvert^2$
Central elastic partial-wave sum	$\sigma_{\mathrm{el}}=(4\pi/k^2)\sum_\ell(2\ell+1)\sin^2\delta_\ell$
Reaction cross section	$\sigma_{\mathrm{reac}}=(\pi/k^2)\sum_\ell(2\ell+1)(1-\lvert S_\ell\rvert^2)$
Inclusive total	$\sigma_{\mathrm{tot}}=\sigma_{\mathrm{el}}+\sigma_{\mathrm{reac}}$
Optical theorem	$\sigma_{\mathrm{tot}}=(4\pi/k)\operatorname{Im}f(0)$
Elastic momentum transfer	$q=2k\sin(\theta/2)$

The partial-wave and optical-theorem rows use the short-range central-potential normalization defined below. Do not combine them with an amplitude from a different $S$- or $T$-matrix convention without translating factors.

Flux definition

For an incident channel $\alpha$, the differential cross section is defined operationally by

$$d\sigma_{\beta\leftarrow\alpha} = \frac{ \text{outgoing rate into channel }\beta \text{ and }d\Omega }{ \text{incident flux in channel }\alpha }.$$

For a scalar Schrödinger wave with reduced mass $\mu$,

$$\mathbf j = \frac{\hbar}{2\mu i} \left( \psi^*\nabla\psi - \psi\nabla\psi^* \right).$$

A unit-amplitude incident plane wave has

$$\mathbf j_{\mathrm{inc}} = \frac{\hbar\mathbf k}{\mu}, \qquad j_{\mathrm{inc}} = \frac{\hbar k}{\mu} = v.$$

For the outgoing elastic spherical wave,

$$j_r^{\mathrm{sc}} \simeq \frac{\hbar k}{\mu} \frac{\lvert f(\theta,\phi)\rvert^2}{r^2}.$$

The rate through the area element

$$dA=r^2d\Omega$$

is $j_r^{\mathrm{sc}}r^2d\Omega$. Dividing by $j_{\mathrm{inc}}$ gives $\lvert f\rvert^2d\Omega$.

The incident–scattered interference current is important in the forward direction and is essential to global flux conservation and the optical theorem. It should not be interpreted as an extra detector-resolved spherical-wave contribution.

See Probability Current and Flux for the full current decomposition.

Dimensions and angular measure

The asymptotic wave fixes

$$[f]=\text{length}.$$

Because solid angle is dimensionless,

$$\left[ \frac{d\sigma}{d\Omega} \right] = [\sigma] = \text{length}^2.$$

In spherical coordinates,

$$d\Omega = \sin\theta\,d\theta\,d\phi.$$

For an azimuthally symmetric amplitude,

$$\sigma = 2\pi \int_0^\pi \sin\theta\, \frac{d\sigma}{d\Omega} \,d\theta.$$

Equivalently, with $x=\cos\theta$,

$$\sigma = 2\pi \int_{-1}^{1} \frac{d\sigma}{d\Omega} \,dx.$$

An experiment usually measures a finite acceptance rather than the full sphere:

$$\sigma_{\mathcal A} = \int_{\mathcal A} d\Omega\, \epsilon(\Omega) \frac{d\sigma}{d\Omega},$$

where $\epsilon(\Omega)$ is the detection efficiency. Partial-wave interference that cancels in a full-sphere integral can survive this restricted integration.

Elastic kinematics and momentum transfer

For elastic scattering in the center-of-mass frame,

$$\lvert\mathbf k_f\rvert = \lvert\mathbf k_i\rvert = k.$$

With

$$\mathbf q=\mathbf k_f-\mathbf k_i,$$

the momentum-transfer magnitude is

$$q = \sqrt{ k^2+k^2-2k^2\cos\theta } = 2k\sin\frac{\theta}{2}.$$

This relation is the bridge between an angular distribution and the Fourier argument used in the Born approximation. If $\hbar\mathbf q$ is called the physical momentum transfer, state that convention explicitly; some texts use $\mathbf q$ for the wave-vector transfer and others for momentum.

Open channels and speed factors

For channel $\gamma$ with threshold energy $\varepsilon_\gamma$, reduced mass $\mu_\gamma$, and total center-of-mass energy $E$,

$$k_\gamma = \frac{ \sqrt{2\mu_\gamma(E-\varepsilon_\gamma)} }{\hbar}, \qquad v_\gamma = \frac{\hbar k_\gamma}{\mu_\gamma}.$$

The channel is open only when $E>\varepsilon_\gamma$. With outgoing amplitudes normalized as spherical-wave coefficients,

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac{ \hbar k_\beta/\mu_\beta }{ \hbar k_\alpha/\mu_\alpha } \lvert f_{\beta\alpha}\rvert^2.$$

If all channels have the same reduced mass, the ratio becomes $k_\beta/k_\alpha$. Closed channels can affect the amplitude virtually but carry no asymptotic outgoing flux and therefore do not appear as detected final channels.

The inclusive cross section from a fixed initial channel is

$$\sigma_{\mathrm{incl},\alpha} = \sum_{\beta\ \mathrm{open}} \int d\Omega\, \frac{v_\beta}{v_\alpha} \lvert f_{\beta\alpha}\rvert^2,$$

with any additional continuous final variables integrated using their appropriate phase-space measure. The channel interpretation is developed at Inelastic Scattering Preview.

Spin sums and averages

If the amplitude carries initial and final spin labels, an unpolarized, incoherent initial ensemble with degeneracy $g_\alpha$ gives

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac{v_\beta}{v_\alpha} \frac{1}{g_\alpha} \sum_{m_\alpha,m_\beta} \left\lvert f_{\beta m_\beta,\alpha m_\alpha} \right\rvert^2.$$

Average over unobserved initial probabilities and sum over orthogonal unobserved final states. Do not average amplitudes across an incoherent mixture. For a coherently prepared superposition, first apply the amplitude matrix to that state and then square.

Polarization-resolved observables retain selected spin labels or density matrices. A formula quoted as “spin averaged” should state which initial states were averaged and which final states were summed.

Central-potential partial waves

For distinguishable spinless particles and a short-range central interaction,

$$f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^{\infty} (2\ell+1) \bigl(S_\ell-1\bigr) P_\ell(\cos\theta).$$

Define

$$a_\ell = \frac{S_\ell-1}{2i}.$$

Then

$$f(\theta) = \frac{1}{k} \sum_{\ell=0}^{\infty} (2\ell+1) a_\ell P_\ell(\cos\theta).$$

For a purely elastic channel,

$$S_\ell=e^{2i\delta_\ell}, \qquad a_\ell=e^{i\delta_\ell}\sin\delta_\ell.$$

At a fixed angle, partial waves add coherently before squaring:

$$\frac{d\sigma}{d\Omega} = \frac{1}{k^2} \left\lvert \sum_{\ell=0}^{\infty} (2\ell+1) a_\ell P_\ell(\cos\theta) \right\rvert^2.$$

Only a complete angular integral removes cross terms by Legendre orthogonality. It yields

$$\sigma_{\mathrm{el}} = \frac{4\pi}{k^2} \sum_{\ell=0}^{\infty} (2\ell+1) \lvert a_\ell\rvert^2,$$

or, in the elastic phase-shift form,

$$\sigma_{\mathrm{el}} = \frac{4\pi}{k^2} \sum_{\ell=0}^{\infty} (2\ell+1) \sin^2\delta_\ell.$$

The complete derivation and convergence diagnostics are at Partial-Wave Cross Sections.

Elastic, reaction, and total cross sections

When other channels are open, write the diagonal elastic element as

$$S_\ell = \eta_\ell e^{2i\delta_\ell}, \qquad 0\le\eta_\ell\le1.$$

The elastic flux returning to the observed channel gives

$$\sigma_{\mathrm{el}} = \frac{\pi}{k^2} \sum_{\ell=0}^{\infty} (2\ell+1) \lvert S_\ell-1\rvert^2.$$

Loss from that channel into all other open channels is the reaction cross section

$$\sigma_{\mathrm{reac}} = \frac{\pi}{k^2} \sum_{\ell=0}^{\infty} (2\ell+1) \left( 1-\lvert S_\ell\rvert^2 \right).$$

The inclusive total is

$$\begin{aligned} \sigma_{\mathrm{tot}} &= \sigma_{\mathrm{el}} + \sigma_{\mathrm{reac}} \\ &= \frac{2\pi}{k^2} \sum_{\ell=0}^{\infty} (2\ell+1) \left( 1-\operatorname{Re}S_\ell \right). \end{aligned}$$

If elastic scattering is the only open process, $\eta_\ell=1$ and $\sigma_{\mathrm{tot}}=\sigma_{\mathrm{el}}$. Once inelastic or absorptive channels exist, calling the elastic integral “the total cross section” is incorrect.

With the amplitude convention above, unitarity also gives

$$\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im}f(0).$$

The forward-amplitude derivation and its multichannel meaning belong to the Optical Theorem Formula Card.

Partial-wave bounds and threshold check

For a purely elastic partial wave,

$$\lvert a_\ell\rvert = \lvert\sin\delta_\ell\rvert \le1.$$

Therefore

$$\sigma_\ell^{\mathrm{el}} \le \frac{4\pi}{k^2}(2\ell+1).$$

The bound is saturated at $\delta_\ell=\pi/2$ modulo $\pi$. It is a maximum allowed by unitarity, not a prediction that every partial wave reaches that value.

For a sufficiently short-range interaction away from threshold anomalies,

$$\delta_\ell=O\left(k^{2\ell+1}\right),$$

so

$$\sigma_\ell^{\mathrm{el}}=O\left(k^{4\ell}\right).$$

The $s$ wave normally dominates at low energy. With scattering length $a_s$,

$$\sigma_{\mathrm{el}} \longrightarrow 4\pi a_s^2$$

for distinguishable particles when the $s$-wave approximation is valid and $k\lvert a_s\rvert\ll1$. Identical-particle symmetry and threshold poles can change this simple limit.

Identical final particles

For two identical spinless particles in a definite spatial-symmetry channel, the direct and exchange alternatives lead to

$$f_\pm(\theta) = f(\theta) \mathbin{\pm} f(\pi-\theta).$$

The sign is fixed by the symmetry of the complete internal and spatial state. The unordered final pair must be counted once. Two equivalent conventions are

$$\sigma_\pm = \int_H d\Omega\, \lvert f_\pm(\theta)\rvert^2$$

over one representative hemisphere $H$, or

$$\sigma_\pm = \frac12 \int_{4\pi}d\Omega\, \lvert f_\pm(\theta)\rvert^2.$$

Do not use both a hemisphere and an additional factor of $1/2$. For spinful particles, combine amplitudes within a coherent spin channel and average probabilities across an incoherent spin ensemble. See Identical-Particle Scattering for even/odd partial-wave selection and spin weights.

Long-range and lower-dimensional caveats

The asymptotic decomposition into an undistorted plane wave plus $f e^{ikr}/r$ assumes sufficiently short-range interactions. For the unscreened Coulomb potential, the incident and outgoing phases contain logarithmic distortions. The Rutherford differential cross section is well-defined at nonzero angle, but its ideal full-angle integral diverges in the forward direction. Use Coulomb asymptotics, screening, wave packets, or a finite experimental angular cutoff as appropriate.

In one spatial dimension, reflection and transmission are dimensionless current ratios:

$$R = \frac{\lvert j_{\mathrm{ref}}\rvert}{j_{\mathrm{inc}}}, \qquad T = \frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}}.$$

They are not three-dimensional cross sections and should not be assigned area units.

From cross section to event rate

For integrated luminosity rate $\mathcal L$ with dimensions $1/(\text{area}\cdot\text{time})$,

$$\frac{dR}{d\Omega} = \mathcal L \frac{d\sigma}{d\Omega}, \qquad R_{\mathcal A} = \mathcal L\sigma_{\mathcal A}.$$

Real event predictions may also require target thickness, beam-energy averaging, branching fractions, detector resolution, efficiency, and background subtraction. A theoretical full-sphere cross section should not be compared directly with uncorrected finite-acceptance counts.

Calculation workflow

1. State the asymptotic normalization of the scattering amplitude.
2. Identify the incident channel, all measured final labels, and every open unobserved channel.
3. Compute incident and outgoing speeds; retain $v_\beta/v_\alpha$ when they differ.
4. Sum coherent amplitudes for indistinguishable alternatives, then square.
5. Average over incoherent initial labels and sum over orthogonal final labels.
6. Integrate over the stated angular acceptance with the correct Jacobian and efficiency.
7. Distinguish elastic, reaction, and inclusive total cross sections.
8. Check dimensions, positivity, partial-wave convergence, and unitarity.
9. Apply identical-particle and long-range modifications before comparing with data.

Common mistakes

• Treating the amplitude $f$ itself as a cross section.
• Omitting $v_\beta/v_\alpha$ in an inelastic channel.
• Mixing a $T$-matrix convention with an incompatible formula for $f$.
• Forgetting $\sin\theta$ in the solid-angle measure.
• Dropping partial-wave interference in a differential or acceptance-limited observable.
• Calling $\sigma_{\mathrm{el}}$ the total cross section when other channels are open.
• Summing incoherent spin or channel amplitudes before squaring.
• Adding probabilities for indistinguishable direct and exchange alternatives.
• Double counting identical final pairs.
• Applying short-range asymptotics to unscreened Coulomb scattering.
• Treating one-dimensional transmission probabilities as area-valued cross sections.
• Comparing a full-theory cross section with detector counts before folding in luminosity and acceptance.

Exercises

1. An elastic amplitude is isotropic, $f(\theta,\phi)=a$. Find the differential and total cross sections and check their dimensions.

Solution

The differential cross section is

$$\frac{d\sigma}{d\Omega} = \lvert a\rvert^2.$$

Integration over the full sphere gives

$$\sigma_{\mathrm{el}} = 4\pi\lvert a\rvert^2.$$

Because $a$ has dimensions of length, both $\lvert a\rvert^2$ and $\sigma_{\mathrm{el}}$ have dimensions of area.

2. A transition has $k_\beta=k_\alpha/2$ and equal reduced masses in the two channels. If $f_{\beta\alpha}=b$ is isotropic, compute its integrated cross section.

Solution

Equal reduced masses imply

$$\frac{v_\beta}{v_\alpha} = \frac{k_\beta}{k_\alpha} = \frac12.$$

Therefore

$$\frac{d\sigma_{\beta\leftarrow\alpha}}{d\Omega} = \frac12\lvert b\rvert^2,$$

and

$$\sigma_{\beta\leftarrow\alpha} = 4\pi\frac12\lvert b\rvert^2 = 2\pi\lvert b\rvert^2.$$

Using $\lvert b\rvert^2$ alone would overestimate the outgoing flux by a factor of two.

3. Suppose only one elastic partial wave $\ell$ is nonzero and its phase shift is $\delta_\ell$. Find its integrated cross section and the value of $\delta_\ell$ that saturates the elastic unitarity limit.

Solution

The partial cross section is

$$\sigma_\ell^{\mathrm{el}} = \frac{4\pi}{k^2} (2\ell+1) \sin^2\delta_\ell.$$

Since $\sin^2\delta_\ell\le1$,

$$\sigma_\ell^{\mathrm{el}} \le \frac{4\pi}{k^2}(2\ell+1).$$

The maximum occurs when

$$\delta_\ell = \frac{\pi}{2} \pmod{\pi}.$$

4. Let the distinguishable-particle amplitude be isotropic, $f(\theta)=a$. Find the event cross sections for symmetric and antisymmetric spatial states of two identical spinless particles.

Solution

The exchange amplitudes are

$$f_+=a+a=2a, \qquad f_-=a-a=0.$$

Integrating once over a hemisphere of solid angle $2\pi$ gives

$$\sigma_+ = 2\pi\lvert2a\rvert^2 = 8\pi\lvert a\rvert^2, \qquad \sigma_-=0.$$

The same result follows from a full-sphere integral multiplied by $1/2$. The symmetric result is twice the distinguishable-particle total $4\pi\lvert a\rvert^2$, while the antisymmetric $s$ wave is forbidden.

Canonical links

• Scattering Amplitude
• Differential and Total Cross Sections
• Probability Current and Flux
• Partial-Wave Cross Sections
• Identical-Particle Scattering
• Inelastic Scattering Preview

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006, Chs. 3, 4, and 11.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982, Chs. 5 and 10.
• C. J. Joachain, Quantum Collision Theory, 3rd ed., North-Holland, 1983, Chs. 3 and 7.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981, Secs. 132–134.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, Ch. 7.
• M. L. Goldberger and K. M. Watson, Collision Theory, Wiley, 1964, Chs. 3–4.