Feshbach Resonance Scattering Length Calculator
This calculator estimates how the s-wave scattering length changes near a magnetically tuned Feshbach resonance and then turns that scattering length into the corresponding zero-energy elastic cross-section. It is aimed at AMO and ultracold-atom work where the background scattering length, resonance width, and resonance position are already known from experiment or a resonance table.
For a single isolated Feshbach resonance, the magnetic-field dependence of the scattering length is modeled by
The symbols mean:
- 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, also in Gauss.
Near the pole at B0, the scattering length can swing through very large positive or negative values, which is why a narrow field scan around a known resonance often matters more than the absolute background value alone.
At ultralow collision energy, the elastic cross-section associated with s-wave scattering is
Here σ is the total elastic cross-section and a is the scattering length expressed in physical units such as meters. Because the input scattering length is entered in Bohr radii a0, the calculator converts it to SI units before evaluating σ.
Using the Feshbach resonance inputs
To evaluate a Feshbach-resonance scattering length, enter the four values that define the single-resonance model:
- Background scattering length abg (a0) – the off-resonant s-wave scattering length, usually taken from experimental measurements or published resonance tables.
- Resonance width Δ (G) – the magnetic width of the resonance in Gauss. This controls how quickly a(B) bends as the field approaches B0.
- Resonance position B0 (G) – the magnetic field at which the resonance occurs for the channel you are studying.
- Magnetic field B (G) – the external field where you want the scattering length and cross-section evaluated.
Use the units shown next to each label: a0 for the background scattering length and Gauss for the magnetic fields. 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 Feshbach-resonance results
The sign and magnitude of the scattering length tell you how the Feshbach resonance is reshaping the interaction between atoms:
- a(B) > 0 and large – the model predicts effectively repulsive interactions and a weakly bound molecular state just below threshold. In that limit, the binding energy follows Eb = ℏ2 / (2 μ a2), where μ is the reduced mass.
- a(B) < 0 and large in magnitude – the interaction is effectively attractive and the entrance channel does not support a true two-body bound state.
- a(B) ≈ 0 – the system is close to a noninteracting point, so elastic collisions are strongly suppressed.
Because the zero-energy cross-section scales as a(B)2, the field detuning from B0 often matters more than the sign by itself when you are checking how strongly the gas will scatter. Very close to the resonance pole, the simple model can return extremely large values, but real experiments are limited by finite temperature, finite collision energy, and inelastic loss channels. Treat those near-pole numbers as a sign that the gas is entering a strongly interacting regime, not as a literal prediction that a cross-section will diverge in the lab.
Worked example: estimating a Feshbach resonance at 805 G
Here is a concrete Feshbach-resonance example with parameters chosen so the algebra is easy to follow:
- abg = 100 a0
- Δ = 10 G
- B0 = 800 G
Suppose you want the scattering properties at B = 805 G. The field detuning is B - B0 = 5 G, so the calculator applies
a(B) = abg [ 1 - Δ / (B - B0) ] = 100 a0 [ 1 - 10 / (805 - 800) ].
Since the denominator is 5 G, the bracket becomes 1 - 2, and the scattering length is
a(B) = 100 a0 [ 1 - 2 ] = -100 a0.
This example shows a common Feshbach-resonance pattern: a modest change in field can flip the sign of the scattering length while leaving its magnitude similar to the background value. The cross-section is then computed from the same scattering length through σ = 4 π a2, with the calculator converting a from Bohr radii to meters before reporting σ in SI units.
If you move the magnetic field to the other side of the pole, the same resonance can produce a very different interaction sign and a much larger magnitude. That is why it is useful to scan several B values around B0 rather than relying on a single field point.
Scattering-length regimes across a Feshbach resonance
The table below summarizes the way a Feshbach resonance typically changes the scattering length as you move the magnetic field from one side of the pole to the other.
| 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 of the single-resonance model
This calculator intentionally uses the simplest standard Feshbach-resonance model. When you apply the results to a real ultracold-gas setup, keep the following points in mind:
- Single, isolated resonance – the expression assumes one resonance with no important overlap from nearby poles. If your species or spin channel has a crowded resonance spectrum, a coupled-channel or multi-resonance treatment is more appropriate.
- Zero-energy, s-wave approximation – only s-wave scattering in the zero-collision-energy limit is included. At higher temperatures, or for higher partial waves, the field dependence and cross-section can differ from the simple formulas shown here.
- Elastic scattering only – the calculator does not model inelastic processes such as spin relaxation, molecule formation loss, or three-body recombination. In an actual experiment, those channels can become important near a resonance even when the elastic cross-section is large.
- Idealized divergence near B0 – mathematically, a(B) diverges as B approaches B0. Experimentally, finite temperature, finite density, and unitarity constraints keep the observable response finite. Extremely large values from the calculator should be read as a warning that the gas is entering a strongly interacting region.
- Inputs must come from data – the calculator does not infer abg, Δ, or B0 on its own. Use reliable published values or measurements for the specific atomic species, hyperfine channel, and resonance you care about.
Within those limits, the tool is useful for quickly checking how a Feshbach resonance reshapes the s-wave scattering length and the zero-energy cross-section, for comparing magnetic-field points on either side of the pole, and for building intuition about strongly interacting ultracold gases.
Formula: how the Feshbach-resonance estimate is assembled
The calculator first evaluates the magnetic-field dependent scattering length a(B) from the single-resonance expression shown above, then inserts that result into σ = 4πa² to obtain the zero-energy elastic cross-section. In practice, the detuning B - B0 is the quantity that drives the sharpest changes: it controls the sign flip, the size of the resonance enhancement, and how quickly the output moves away from the background value.
That is why this calculator is most useful for field scans, resonance checks, and quick order-of-magnitude estimates when you already know the underlying resonance parameters. If the chosen field sits very close to the pole, the output should be read as a strong-interaction indicator rather than a literal promise of an infinite laboratory cross-section.
Arcade Mini-Game: Feshbach Resonance Input Check
Use this quick arcade run to practice spotting the background scattering length, resonance width, and resonance position that drive a Feshbach-resonance calculation.
Start the game, then use your pointer or arrow keys to catch the resonance inputs and avoid distracting assumptions.