Scattering Length

The scattering length $a$ is the leading threshold parameter for $s$-wave scattering from a short-range potential. It determines the low-energy amplitude

$$f_0(k) \approx - \frac{a}{1+ika}$$

when effective-range corrections can be neglected.

Despite its name, $a$ is not always the literal size of the potential. It can be much larger than the range, negative, positive, or divergent when a bound state sits near threshold.

Definition from the Phase Shift

For $s$-wave scattering, define

$$a = - \lim_{k\to0} \frac{\tan\delta_0(k)}{k}.$$

Equivalently,

$$k\cot\delta_0(k) = - \frac1a +O(k^2).$$

With this convention, a hard sphere of radius $R$ has

$$a=R.$$

Different sign conventions exist in older literature, so always check the relation between $a$ and $\delta_0$ before comparing formulas.

Zero-Energy Wavefunction Definition

At zero energy and outside the range of a short-range potential, the $s$-wave radial equation gives a linear radial solution:

$$u_0(r) \propto r-a.$$

Equivalently, the full radial wavefunction behaves as

$$\psi(r) \propto 1-\frac{a}{r}.$$

Thus $a$ is the intercept obtained by extending the zero-energy outside solution back to the radial axis.

This geometric picture is useful, but it should not be overinterpreted. The intercept can lie far outside the potential range when the system is near a threshold pole.

Low-Energy Amplitude

The exact $s$-wave amplitude can be written as

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

Using

$$k\cot\delta_0(k) \approx - \frac1a$$

gives

$$f_0(k) \approx - \frac{a}{1+ika}.$$

The corresponding cross section for distinguishable particles is

$$\sigma \approx \frac{4\pi a^2}{1+k^2a^2}.$$

At very low energy with $k|a|\ll1$,

$$\sigma\approx4\pi a^2.$$

Sign Conventions and Interpretation

The sign alone does not determine whether an interaction is attractive or whether it binds. A hard sphere has $a>0$ and no bound state. For a regular short-range interaction with $a$ large and positive compared with its range and $|r_e|\ll a$, a shallow bound-state pole has positive binding energy

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

A negative scattering length often indicates attractive interaction without a physical shallow bound state, or a nearby virtual state. This statement is qualitative unless the analytic structure of the $S$-matrix is specified.

Small positive $a$ does not necessarily imply a shallow universal bound state; the universal formula requires $a$ to be large compared with the range.

Hard-Sphere Example

For an impenetrable sphere of radius $R$, the radial wavefunction must vanish at $r=R$:

$$u_0(R)=0.$$

The outside zero-energy solution is proportional to $r-a$. Applying the boundary condition gives

$$R-a=0,$$

so

$$a=R.$$

The hard sphere is one of the few cases where the scattering length is literally the obstacle radius. Hard-Sphere Scattering derives the exact finite-energy phases, also obtains $r_e=2R/3$, and explains why the threshold cross section is four times the geometric area.

Attractive Square Well

For an attractive spherical square well

$$V(r) = \begin{cases} -V_0, & r<R,\\ 0, & r>R, \end{cases} \qquad V_0>0,$$

the zero-energy inside solution is

$$u_{\mathrm{in}}(r) = A\sin(\kappa r), \qquad \kappa = \frac{\sqrt{2\mu V_0}}{\hbar}.$$

Outside,

$$u_{\mathrm{out}}(r) = B(r-a).$$

Matching logarithmic derivatives at $r=R$ gives

$$a = R - \frac{\tan(\kappa R)}{\kappa}.$$

The scattering length diverges when $\tan(\kappa R)$ diverges. Those divergences occur when a new $s$-wave bound state appears at threshold.

Validity of the Born Approximation compares this exact result with its smooth first Born expansion and uses the first divergence to diagnose nonperturbative threshold scattering. Square-Well Scattering embeds the zero-energy result in the exact finite-energy partial-wave solution and contrasts the $s$-wave threshold pole with a $p$-wave shape resonance.

Large Scattering Length

When

$$|a|\gg R,$$

the system is near the unitary regime. The amplitude is then controlled by the pole structure rather than by the microscopic range. In the formal limit $|a|\to\infty$,

$$f_0(k)\to\frac{i}{k}, \qquad \sigma_0\to\frac{4\pi}{k^2}.$$

This limit also requires $k|r_e|\ll1$ and negligible higher shape terms. For finite large $|a|$, it describes the window $1/|a|\ll k\ll1/R$, not the strict $k\to0$ limit. It is the $s$-wave unitarity limit, consistent with probability conservation.

Common Mistakes

• Interpreting $a$ as the physical radius of the potential in all cases.
• Forgetting the minus sign in $a=-\lim_{k\to0}\tan\delta_0/k$.
• Assuming positive $a$ always guarantees a universal shallow bound state.
• Applying scattering-length formulas to Coulomb scattering without screening or a modified formalism.
• Ignoring effective range when $k r_e$ is comparable to one.
• Treating a divergent $a$ as a mathematical accident rather than a threshold pole.

Exercises

1. A zero-energy outside solution has $u(r)\propto r-3R$. What is the scattering length?

Solution

By definition, outside the potential range

$$u(r)\propto r-a.$$

Comparing with $u(r)\propto r-3R$ gives

$$a=3R.$$

2. Show that a hard sphere of radius $R$ has $\delta_0\approx-kR$ at low energy and hence $a=R$.

Solution

Outside the hard sphere,

$$u_0(r) \propto \sin(kr+\delta_0).$$

The boundary condition $u_0(R)=0$ gives

$$kR+\delta_0=0$$

modulo $\pi$. The low-energy branch is

$$\delta_0\approx-kR.$$

Then

$$a = - \lim_{k\to0} \frac{\tan\delta_0}{k} = - \lim_{k\to0} \frac{-kR}{k} =R.$$

3. Why does the square-well scattering length diverge when a new bound state reaches threshold?

Solution

At threshold, the outside zero-energy solution becomes extremely extended. In the formula

$$a = R - \frac{\tan(\kappa R)}{\kappa},$$

divergence occurs when $\tan(\kappa R)$ diverges. These are precisely the parameter values where the zero-energy matching condition supports a state at threshold. Moving through such a value changes the number of bound states.

References

• H. A. Bethe, “Theory of the Effective Range in Nuclear Scattering,” Physical Review 76, 38–50 (1949), doi:10.1103/PhysRev.76.38.
• C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, “Feshbach Resonances in Ultracold Gases,” Reviews of Modern Physics 82, 1225–1286 (2010), doi:10.1103/RevModPhys.82.1225.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.