Coulomb Scattering

Coulomb scattering is the central example of long-range scattering. The potential

$$V(r)=\frac{\kappa}{r}$$

falls too slowly for the standard short-range asymptotic form to apply without modification. The differential cross section has the Rutherford form, but the scattering state carries long-range logarithmic phase structure.

This page treats nonrelativistic two-body Coulomb scattering in the center-of-mass frame, with reduced mass $\mu$ and relative momentum $\hbar k$. For the place of positive-energy states in the Coulomb spectrum before scattering amplitudes are introduced, see Continuum States of the Coulomb Problem.

Why Coulomb Is Special

For a short-range potential, the large-distance scattering state has the schematic form

$$\psi(\mathbf r) \sim e^{ikz} + f(\theta) \frac{e^{ikr}}{r}.$$

The Coulomb potential does not become negligible fast enough for the incident wave to remain a simple plane wave at large distance. Instead, both incoming and outgoing waves acquire Coulomb phases. This affects:

• the definition of asymptotic states,
• the partial-wave phase shifts,
• the forward-angle behavior,
• the total cross section,
• the use of the optical theorem.

The familiar Rutherford cross section is correct, but it sits inside a more subtle scattering theory than short-range potential scattering.

Coulomb Parameter

Define the dimensionless Coulomb parameter

$$\eta = \frac{\mu\kappa}{\hbar^2k}.$$

Repulsive Coulomb scattering has $\kappa>0$, while attractive Coulomb scattering has $\kappa<0$. The energy is

$$E = \frac{\hbar^2k^2}{2\mu}.$$

The magnitude $\lvert\eta\rvert$ grows at low energy, so Coulomb scattering is not a small low-energy perturbation.

Rutherford Cross Section

The exact Coulomb differential cross section is

$$\frac{d\sigma}{d\Omega} = \frac{\eta^2}{4k^2\sin^4(\theta/2)}.$$

Equivalently,

$$\frac{d\sigma}{d\Omega} = \frac{\kappa^2}{16E^2\sin^4(\theta/2)}.$$

This is the Rutherford formula. It agrees with the classical result for a $1/r$ force.

The forward divergence as $\theta\to0$ reflects the long range of the force. The total cross section obtained by integrating over all angles diverges unless screening, finite beam geometry, or an angular cutoff is included.

Exact Amplitude and Phase

One common convention for the Coulomb amplitude is

$$f_C(\theta) = - \frac{\eta}{2k\sin^2(\theta/2)} \exp \left[ 2i\sigma_0 - 2i\eta\ln\sin\frac{\theta}{2} \right],$$

where

$$\sigma_\ell = \arg\Gamma(\ell+1+i\eta)$$

is the Coulomb phase shift. The phase convention may differ between texts, but the magnitude gives the Rutherford cross section:

$$\lvert f_C(\theta)\rvert^2 = \frac{\eta^2}{4k^2\sin^4(\theta/2)}.$$

The logarithm in the phase is a signature of the long-range tail. It has no analogue for ordinary finite-range potentials.

Deriving the logarithmic phases

The phases follow from an exact scattering state, rather than from a change to the Rutherford magnitude. Take the incident direction along $+z$, retain the time convention $e^{-iEt/\hbar}$, and define

$$s=k(r-z)=2kr\sin^2\frac{\theta}{2}, \qquad \zeta=is.$$

Substituting $\psi=e^{ikz}F(\zeta)$ into $[\nabla^2+k^2-2\eta k/r]\psi=0$ gives

$$\zeta F''+(1-\zeta)F'+i\eta F=0.$$

This is Kummer’s equation with parameters $a=-i\eta$ and $b=1$. Its regular solution, normalized to give unit coefficient for the incident branch, is

$$\psi_C^{(+)}(\mathbf r) =e^{-\pi\eta/2}\Gamma(1+i\eta)e^{ikz} \,{}_1F_1(-i\eta;1;is).$$

At fixed $\eta$ and fixed nonzero angle, the two large-$s$ branches of the confluent hypergeometric function yield

$$\begin{aligned} \psi_C^{(+)}(\mathbf r) ={}&e^{ikz+i\eta\ln s} \left[1-\frac{i\eta^2}{s}+O(s^{-2})\right]\\ &+\frac{f_C(\theta)}{r} e^{ikr-i\eta\ln(2kr)} \left[1+O(s^{-1})\right]. \end{aligned}$$

The gamma-function identities $\Gamma(1-i\eta)=(-i\eta)\Gamma(-i\eta)$ and $\Gamma(1+i\eta)/\Gamma(1-i\eta)=e^{2i\sigma_0}$ give the coefficient $f_C$ displayed above. Thus the incident branch has the phase $+\eta\ln[k(r-z)]$, while the outgoing branch has $-\eta\ln(2kr)$. Both follow from the same exact wave. The $1/s$ incident correction is included because, at fixed angle, it is also of order $1/r$; dropping it while trying to extract the outgoing coefficient from an exact wave would contaminate that coefficient. See the NIST Kummer-function formulas and Pijnenburg et al. (2022), Eqs. (18)–(25).

This separation is not uniform in the forward direction: $s$ need not be large when $\theta$ approaches zero as $r$ grows. Indeed, on the forward axis ${}_1F_1(-i\eta;1;0)=1$, so the exact stationary wave is finite at finite $r$, even though the fixed-angle amplitude diverges as $\theta\to0$. These statements concern different limits; the exact wave does not make the ideal Rutherford angular integral finite. For $\eta\to0$ the exact state reduces to $e^{ikz}$.

Partial Waves

For Coulomb scattering, the partial-wave $S$-matrix contains Coulomb phases

$$S_\ell^{C} = e^{2i\sigma_\ell}.$$

The regular radial Coulomb function has the large-radius behavior

$$F_\ell(\eta,kr)\sim \sin\!\left(kr-\frac{\ell\pi}{2}-\eta\ln(2kr)+\sigma_\ell\right).$$

The extra $-\eta\ln(2kr)$ grows with radius. It cannot be absorbed into a constant short-range phase shift. The sign follows the potential $+\kappa/r$ and time dependence $e^{-iEt/\hbar}$ used here. Unlike short-range scattering, infinitely many partial waves contribute because the interaction reaches arbitrarily large impact parameters.

When a short-range nuclear or atomic potential is added to a Coulomb interaction, one often separates the known Coulomb phases from additional short-range phase shifts:

$$S_\ell = e^{2i\sigma_\ell} e^{2i\delta_\ell^{\mathrm{short}}}.$$

This separation is essential in charged-particle scattering.

Born Approximation Subtlety

Let $\mathbf q=\mathbf k'-\mathbf k$ be the wave-vector transfer; the projectile momentum change is $\hbar\mathbf q$. Formally, the Fourier transform of the Coulomb potential is

$$\int d^3r\, e^{-i\mathbf q\cdot\mathbf r} \frac{\kappa}{r} = \frac{4\pi\kappa}{q^2}.$$

Substituting this into the first Born formula gives

$$f_{\mathrm B}(q) = - \frac{2\mu\kappa}{\hbar^2q^2}.$$

For elastic scattering,

$$q=2k\sin\frac{\theta}{2},$$

so

$$\lvert f_{\mathrm B}\rvert^2 = \frac{\eta^2}{4k^2\sin^4(\theta/2)}.$$

The magnitude matches the Rutherford cross section. This agreement is special. It does not mean the unscreened Coulomb problem is a well-behaved ordinary Born approximation: the exact phase and asymptotic states are long-range modified, and the forward divergence remains. Validity of the Born Approximation explains why magnitude agreement does not establish convergence and compares short-range and Coulomb diagnostics.

Screening and Physical Cutoffs

Real scattering experiments rarely involve an ideal infinite Coulomb field with perfect angular resolution. Atomic electrons screen nuclear charge at long distances. Beams have finite size. Detectors exclude a small forward cone. Plasmas have Debye screening.

A screened Coulomb potential behaves like a short-range potential beyond the screening length. One may compute with screening and then take controlled limits for observables that remain finite. Inclusive or cutoff-defined observables are often more physical than a formal total cross section.

Yukawa Potential in the Born Approximation performs this calculation explicitly and shows that removing the screening makes the total cross section diverge as the inverse screening scale squared while the momentum-transfer cross section diverges only logarithmically.

Exercises

1. Starting from the Born amplitude

$$f_{\mathrm B}(q) = - \frac{2\mu\kappa}{\hbar^2q^2},$$

derive the Rutherford angular dependence.

Solution

For elastic scattering,

$$q=2k\sin\frac{\theta}{2}.$$

Therefore

$$f_{\mathrm B} = - \frac{2\mu\kappa}{\hbar^2} \frac{1}{4k^2\sin^2(\theta/2)} = - \frac{\eta}{2k\sin^2(\theta/2)}.$$

Taking the magnitude squared gives

$$\frac{d\sigma}{d\Omega} = \lvert f_{\mathrm B}\rvert^2 = \frac{\eta^2}{4k^2\sin^4(\theta/2)}.$$

2. Why does the total Coulomb cross section diverge?

Solution

Near $\theta=0$,

$$\sin\frac{\theta}{2}\approx\frac{\theta}{2},$$

so

$$\frac{d\sigma}{d\Omega} \propto \frac{1}{\theta^4}.$$

The solid-angle element behaves as $d\Omega\approx2\pi\theta\,d\theta$, so the forward integral contains

$$\int^{ } \frac{\theta\,d\theta}{\theta^4} = \int^{ } \frac{d\theta}{\theta^3},$$

which diverges at the lower limit. For a detector accepting $\theta_{\min}\le\theta\le\pi$, integration with $u=\sin(\theta/2)$ gives the finite result

$$\sigma(\theta\ge\theta_{\min}) =\frac{\pi\eta^2}{k^2}\cot^2\frac{\theta_{\min}}{2}.$$

This is a cross section for the stated angular acceptance. Its divergence as the excluded cone shrinks shows explicitly why an ideal unscreened total is infinite.

3. Explain why the agreement between the Born magnitude and the Rutherford cross section does not make Coulomb scattering an ordinary short-range Born problem.

Solution

The Coulomb potential is long-ranged, so the standard plane-wave plus outgoing spherical-wave asymptotic form is modified by Coulomb phases. The exact amplitude contains a logarithmic angular phase and Coulomb phase shifts. The Born expression reproduces the magnitude of the Rutherford cross section, but it does not reproduce the full long-range phase structure or remove the forward divergence.

References

• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• F. W. J. Olver et al., eds., NIST Digital Library of Mathematical Functions, §13.2, Kummer’s equation and functions, and §13.7, large-argument expansions, especially Eqs. 13.2.1 and 13.7.2.
• M. Pijnenburg, G. Cusin, C. Pitrou, and J.-P. Uzan, “Rutherford scattering of quantum and classical fields,” (2022), arXiv:2211.14113, §§III A–B.
• E. Rutherford, “The scattering of alpha and beta particles by matter and the structure of the atom,” Philosophical Magazine 21, 669–688 (1911), doi:10.1080/14786440508637080.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.