Low-energy two-body scattering

Numerical laboratory for the low-energy limit of two-body scattering, built as the first stage of my MSc at the São Carlos Institute of Physics (IFSC-USP).

Universality. At low energy only two numbers survive, the scattering length a and the effective range r0. The same two-parameter description then holds for systems with nothing else in common: the deuteron, bound by the strong force at the MeV scale, and the helium-4 dimer, bound by van der Waals at the mK scale. Converted to the same units, that is 13 orders of magnitude in energy and 6 in length.

What is measured here. Both systems, from (a, r0) alone, with four unrelated potentials each. The binding energies come out within 1% of the published values, and the four potentials agree with one another to the same 1% — the shape-independent approximation, Eq. (55) of the reference. Their tuned parameters are of course all different; only (a, r0) is shared.

This work reproduces Macêdo-Lima & Madeira, Rev. Bras. Ensino Fís. 45, e20230079 (2023), and checks itself against the closed forms published there.

Contents

notebooks/1_um_corpo/        one particle in a central potential
    schrodinger.py           harmonic oscillator, hydrogen, box
    one_body.ipynb

notebooks/2_dois_corpos/     the two-body laboratory
    lab.py                   potentials, Numerov solver, tabulated data
    test_lab.py              18 checks against closed forms and published tables
    two_body_scattering.ipynb

notas_teoria/
    Theory_and_Implementation.pdf every function and every constant explained

Each notebook needs only the .py file sitting next to it. Nothing else.

Running it

pip install -r requirements.txt
jupyter lab notebooks/2_dois_corpos/two_body_scattering.ipynb   # about 2 min, Run All
python notebooks/2_dois_corpos/test_lab.py                      # the checks

How it is verified

Every check compares against something the code did not compute: a closed form, an analytic condition, or a published table.

check	anchor	agreement
a of the square well	Eq. (80) of the article	2e-10
r0 of the square well	Eq. (92) of the article	6e-12
bound state	k cot(kR) = -kappa	3e-10
r0/R at the poles of a	exact prediction, Fig. 6	1.0000000000
node counts, 12 cases	Table 2	exact
tuning, 12 cases	Tables 3 and 4	9 within 0.2%
helium dimer, 4 potentials	Motovilov et al., Table I	see below
each Aziz minimum	its own (rm, -eps)	1e-9
the grid ends exactly on R	120 reduced masses	exact
physics under r -> r/L	four rescalings	1e-12
every number since the last run	recorded values	1e-12
grid convergence	halving the points	no change beyond 1e-6

The helium row is the one that costs nothing and proves the most, because no parameter in it was fitted here. Published parameters go in, published numbers come out:

potential	a (Å)	published	E (mK)	published
HFDHE2	124.65	124.65	−0.83012	−0.83012
HFD-B	88.60	88.50	−1.68541	−1.68541
LM2M2	100.23	100.23	−1.30348	−1.30348
TTY	100.01	100.01	−1.30962	−1.30962

Every energy agrees to six figures. HFD-B is the exception on a, 0.11% out while its energy agrees like the others; it is left visible rather than tuned away. Reference: Motovilov, Sandhas, Sofianos & Kolganova, Eur. Phys. J. D 13, 33 (2001), Table I for the results and the Appendix for the parameters.

Two of these checks earned their place by catching something. Grid convergence found the one real bug in the physics: on a uniform grid the Lennard-Jones r0 drifted instead of converging, because its hard core and its tail differ by five orders of magnitude in scale, and a logarithmic grid fixed it. The endpoint check found a silent one: exp(log(R)) does not always return R, so for 14% of reduced masses the last grid point fell outside the potential and the answer was wrong in the seventh figure instead of the tenth.

Author

Pedro Henrique Gesualdo Modesto — São Carlos Institute of Physics, USP. Advisor: Lucas Madeira. Co-advisor: Patrícia C. M. Castilho. Supported by CAPES.