Effective-Range Expansion

The effective-range expansion is the low-energy expansion of the $s$-wave phase shift for short-range interactions. It refines scattering-length universality by adding the first finite-range correction. Low-Energy S-Wave Scattering tests that correction against an exact square-well amplitude and its shallow bound-state pole.

With the phase-shift convention used in this volume,

$$f_0(k) = \frac{1}{k\cot\delta_0(k)-ik}.$$

For many short-range potentials,

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + P k^4 + \cdots.$$

Here $a$ is the scattering length, $r_e$ is the effective range, and $P$ is a shape parameter. This page is the canonical home for this expansion. The leading parameter $a$ is treated in more detail on Scattering Length.

Why Expand k cot delta

The phase shift itself has the low-energy behavior

$$\delta_0(k) \sim - ka$$

when $a$ is finite. The combination $k\cot\delta_0(k)$ is more useful because it appears directly in the denominator of the scattering amplitude:

$$f_0(k) = \frac{1}{k\cot\delta_0(k)-ik}.$$

For short-range potentials, $k\cot\delta_0(k)$ is often analytic in $k^2$ near threshold. This gives an expansion in even powers of $k$.

Scattering Length and Effective Range

Keeping the first two terms gives

$$f_0(k) \approx \frac{1} { -1/a + (r_e/2)k^2 - ik }.$$

The scattering length controls the leading threshold amplitude. The effective range controls the first correction that remembers the finite spatial extent and shape of the interaction.

The dimension of $r_e$ is length. For a natural short-range potential of range $R$, one often expects

$$r_e\sim R.$$

This expectation can fail near fine tuning, in multichannel problems, or when the force has a long tail.

Bound-State Pole Check

A bound-state pole occurs at imaginary wave number

$$k=i\kappa, \qquad \kappa>0,$$

with binding energy

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

The pole condition is that the denominator of $f_0$ vanish:

$$k\cot\delta_0(k)-ik=0.$$

Using the effective-range expansion at $k=i\kappa$ gives

$$- \frac1a - \frac12r_e\kappa^2 + \kappa + \cdots = 0.$$

If $|a|$ is very large and $r_e\kappa\ll1$, this reduces to

$$\kappa\approx\frac1a.$$

For large positive $a$, the shallow binding energy is therefore

$$E_b \approx \frac{\hbar^2}{2\mu a^2}.$$

The effective range gives the first correction to this universal relation.

Analyticity Assumptions

The expansion is not a purely algebraic identity. It relies on threshold analyticity properties of short-range scattering. It can fail or require modification for:

• unscreened Coulomb interactions,
• long-range tails such as polarization potentials,
• singular short-distance potentials,
• nearby inelastic thresholds,
• channels with strong coupling to closed-channel states,
• energies beyond the threshold region.

The phrase “effective range” should therefore be read with the range of validity attached.

Relation to EFT

In low-energy effective field theory, the scattering length is represented by a leading contact interaction and the effective range by derivative corrections. The expansion

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + \cdots$$

is the scattering-theory version of an expansion in powers of momentum over a short-distance scale.

When the magnitude of $a$ is natural, ordinary power counting often works directly. When the magnitude of $a$ is much larger than the range, the leading interaction must be treated nonperturbatively, while effective-range and shape corrections can still be organized systematically.

Common Mistakes

• Using only the scattering length when $k|r_e|$ is not small.
• Applying the expansion to Coulomb scattering without the Coulomb-modified version.
• Assuming $r_e$ is always positive.
• Treating a large $a$ as weak scattering rather than as a sign of a nearby pole.
• Extending the threshold expansion beyond its radius of usefulness.

Exercises

1. Starting from the effective-range amplitude, recover the scattering-length approximation by setting $r_e=0$.

Solution

The effective-range amplitude is

$$f_0(k) \approx \frac{1} { -1/a + (r_e/2)k^2 - ik }.$$

Setting $r_e=0$ gives

$$f_0(k) \approx \frac{1}{-1/a-ik} = - \frac{a}{1+ika},$$

which is the scattering-length approximation.

2. Derive the shallow-bound-state pole equation including $r_e$.

Solution

At a bound-state pole, $k=i\kappa$ with $\kappa>0$. The denominator of $f_0$ vanishes:

$$k\cot\delta_0(k)-ik=0.$$

Using

$$k\cot\delta_0(k) = - \frac1a + \frac12r_ek^2 + \cdots,$$

and $k^2=-\kappa^2$, one gets

$$- \frac1a - \frac12r_e\kappa^2 + \kappa + \cdots =0.$$

3. Why is the expansion written in powers of $k^2$ rather than $k$?

Solution

For ordinary short-range, time-reversal-invariant elastic $s$-wave scattering, the real threshold function $k\cot\delta_0(k)$ is analytic in the energy, and the energy is proportional to $k^2$. This gives an expansion in even powers of $k$. Long-range forces or threshold singularities can change this analytic structure.

References

• H. A. Bethe, “Theory of the effective range in nuclear scattering,” Physical Review 76, 38-50, 1949.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• H.-W. Hammer, S. Konig, and U. van Kolck, “Nuclear effective field theory: status and perspectives,” Reviews of Modern Physics 92, 025004, 2020.