Feshbach Resonance Scattering Length Calculator
This calculator estimates the s-wave scattering length and corresponding zero-energy scattering cross-section for ultracold atomic collisions near a magnetically tuned Feshbach resonance. It is intended for AMO/ultracold-atom researchers and students who already know the relevant resonance parameters from experiment or the literature.
Near a single, isolated Feshbach resonance, the dependence of the scattering length on magnetic field is modeled by
where:
- a(B) is the s-wave scattering length at magnetic field B, typically expressed in Bohr radii a0.
- abg is the background scattering length far from resonance (in units of a0).
- Δ is the magnetic-field width of the resonance (in Gauss).
- B0 is the resonance position (in Gauss).
In the zero-energy limit of ultracold collisions, the elastic cross-section associated with s-wave scattering is
where σ is the total elastic cross-section and a is the scattering length expressed in physical units (e.g., meters). When a is entered in units of Bohr radii a0, the calculator uses the Bohr radius conversion to return a cross-section in SI units.
How to use this calculator
To evaluate the scattering properties near a Feshbach resonance, provide the following inputs:
- Background scattering length abg (a0) – the off-resonant s-wave scattering length, in units of Bohr radii. This is typically taken from experimental measurements or published resonance tables.
- Resonance width Δ (G) – the magnetic width of the Feshbach resonance in Gauss. This parameter characterizes how quickly a(B) varies with B.
- Resonance position B0 (G) – the magnetic field at which the resonance occurs, usually quoted for a specific hyperfine channel.
- Magnetic field B (G) – the external magnetic field at which you want to evaluate the scattering length and cross-section, in Gauss.
All inputs should be provided in the units indicated (a0 and Gauss). The calculator returns:
- the scattering length a(B) in Bohr radii, and
- the corresponding s-wave elastic cross-section σ at zero collision energy.
Interpreting the results
The sign and magnitude of the scattering length carry important physical meaning for ultracold atomic gases:
- a(B) > 0 and large – effectively repulsive interactions and the presence of a weakly bound molecular state just below threshold. The molecular binding energy is approximately Eb = ℏ2 / (2 μ a2), where μ is the reduced mass.
- a(B) < 0 and large in magnitude – effectively attractive interactions with no true two-body bound state in the entrance channel.
- a(B) ≈ 0 – near-noninteracting regime, where elastic collisions are strongly suppressed.
On either side of the resonance, the cross-section grows as a(B)2. Very close to resonance, |a(B)| can become extremely large within this simple model, implying enormous cross-sections. In real systems, unitarity limits at finite collision energy and inelastic channels prevent the cross-section from diverging without bound, so those extreme values should be interpreted with care.
Worked example
As a concrete illustration, consider an alkali-atom Feshbach resonance with the following approximate parameters:
- abg = 100 a0
- Δ = 10 G
- B0 = 800 G
Suppose you want to evaluate the scattering properties at B = 805 G. Using the formula above,
a(B) = abg [ 1 - Δ / (B - B0) ] = 100 a0 [ 1 - 10 / (805 - 800) ].
The denominator is 5 G, so
a(B) = 100 a0 [ 1 - 10 / 5 ] = 100 a0 [ 1 - 2 ] = -100 a0.
The scattering length is negative and has the same magnitude as the background value in this simple example. The corresponding zero-energy cross-section is
σ = 4 π a2 = 4 π (100 a0)2.
Expressed in Bohr radii squared, this is 4 π × 104 a02, which the calculator converts to SI units by multiplying by a02 in square meters.
By varying B and repeating the calculation, you can map out how the interaction strength evolves across the resonance. For example, choosing B slightly below B0 (so that B - B0 is negative) would yield a large positive a(B), corresponding to a weakly bound molecular state and strongly repulsive effective interactions between atoms.
Comparison of regimes
The table below summarizes typical qualitative regimes you may encounter when scanning B across a resonance, in the context of ultracold atomic gases and magnetically tuned Feshbach resonances.
| Magnetic field B | Typical a(B) | Interaction character | Cross-section σ |
|---|---|---|---|
| Far below B0 | a(B) ≈ abg | Weak, background interactions | Modest, set by abg2 |
| Approaching B0 from below | |a(B)| grows, often large and positive | Strongly interacting, molecular state supported | Rapidly increasing until unitarity or losses dominate |
| Very close to B0 | |a(B)| → ∞ in the simple model | Unitary regime in real gases | Formally divergent in the model; physically limited |
| Just above B0 | Large negative a(B) | Effectively attractive, no bound dimer in entrance channel | Large but finite; sensitive to field stability |
| Far above B0 | a(B) &to; abg | Return to background interactions | Again set by abg2 |
Assumptions and limitations
This calculator is deliberately based on the simplest single-resonance Feshbach model. When applying the results to real experiments, keep the following assumptions and limitations in mind:
- Single, isolated resonance – the formula assumes a single Feshbach resonance with no significant overlap with nearby resonances. In species or channels with dense resonance spectra, a multi-resonance or full coupled-channel model is more appropriate.
- Zero-energy, s-wave approximation – only s-wave (l = 0) scattering in the zero-collision-energy limit is included. At higher temperatures or for higher partial waves, additional energy dependence and angular-momentum channels change both a(B) and σ.
- Elastic scattering only – inelastic processes (e.g., three-body recombination, spin relaxation, molecule formation losses) are not modeled. Near resonance, such loss channels can dominate dynamics even when the elastic cross-section is large.
- Idealized divergence – mathematically, a(B) diverges when B → B0. In practice, finite temperature, density-dependent effects, and unitarity limits cap the effective cross-section. Extremely large values returned by this calculator close to B0 should be treated as indicating a strongly interacting regime rather than literal, experimentally achievable cross-sections.
- Input parameters from experiment or literature – the calculator does not determine abg, Δ, or B0. You must obtain these from high-quality measurements or reliable compilations for the specific atomic species, spin states, and channels of interest.
Within these constraints, the tool is useful for quickly exploring how magnetically tuned Feshbach resonances modify the s-wave scattering length and zero-energy cross-section in ultracold atomic gases, for planning experiments, and for building intuition about strongly interacting regimes such as the BEC–BCS crossover or unitary Fermi gases.