Conventions

There are two common conventions for defining the parameters of the effective range expansion. The more common in nuclear physics, which keeps the effective range expansion simple

\[\cot \delta(k) = \frac{2}{\pi} \log(ka_2) + \sigma_2 k^2 + \mathcal{O}(k^4) \text{ analytic in }k\]

where \(a_2\) is the scattering length and \(\sqrt{|\sigma_2|}\) is the effective range [1].

The other convention is from hard-disk and square-well scattering. The scattering length is the radius at which the wavefunction (upon linear extrapolation from the disk’s boundary) hits 0. According to [3, 4, 5, 6] we recover this geometric meaning when

\[\cot \delta(k) = \frac{2}{\pi}\left[\log \frac{ka_{2D}}{2} + \gamma\right] + \frac{1}{4} R_e^2 k^2 + \cdots,\]

\(R_e\) has a similar geometric meaning (see (18) in Ref. [6]), and \(\gamma=0.5772\ldots\) is the Euler-Mascheroni constant.

Note

We use the nuclear physics convention and implement it in the EffectiveRangeExpansion.

Warning

We typically do not write the subscript on \(a\), especially in code! We just assume the nuclear convention.

To help convert from the geometric convention we provide

class tdg.conventions.from_geometric

static scattering_length(geometric)

To match the log terms we must identify

\[a_{2} = \frac{e^{\gamma}}{2} a_{2D} = (0.8905362090\cdots) a_{2D}.\]

static log_ka(geometric)

To match the logs we must identify

\[\log k a_2 = \log k a_{2D} + \gamma - \log 2.\]

static alpha(geometric)

static n_asquared_to_alpha(geometric)

Some references quote the dimensionless combination of the number density and the square of the geometric-convention scattering length \(na_{2D}^2\).

This can be converted to \(\alpha\)