Radioactive Decay Chains, Rutherford Scattering, and Nuclear Binding#

This tutorial covers three classic pieces of nuclear physics with physicskit.particle.decays, physicskit.particle.scattering, and physicskit.particle.nuclear: how a chain of successive radioactive decays evolves, how alpha particles scatter off a nucleus, and why iron sits at the peak of nuclear stability.

A three-species decay chain#

A radioactive chain \(1\to2\to3\) – an unstable parent decaying to an unstable daughter, which itself decays to a stable granddaughter – is governed by the Bateman equations, implemented directly by bateman_decay_chain():

import numpy as np
from physicskit.particle.decays import bateman_decay_chain

t = np.array([0.0, 2.0, 5.0, 10.0, 20.0])
N = bateman_decay_chain(N0=1000.0, decay_constants=[0.5, 0.2, 0.05], t=t)
print(np.round(N, 1))
# [[1000.     367.9     82.1      6.7      0. ]
#  [   0.     504.1    476.3    214.3     30.5]
#  [   0.     123.4    397.1    602.8    504.3]]

Each row is one species’ population over time: the parent (row 0) decays away monotonically; the daughter (row 1) first builds up as it’s fed by the parent, then decays away itself once the parent is nearly gone; the granddaughter (row 2) keeps accumulating population throughout, since it is treated as stable in this example (its own decay constant, 0.05, is just slow on this timescale). Species 1 alone, by construction, matches the plain exponential law radioactive_decay_number() exactly – the whole chain machinery reduces to the simple case when there’s nothing downstream feeding back.

Rutherford scattering off a gold nucleus#

Geiger and Marsden’s classic experiment fired alpha particles (\(Z_1=2\)) at a thin gold foil (\(Z_2=79\)). rutherford_dsigma_domega() reproduces the steep angular falloff Rutherford predicted:

from physicskit.particle.scattering import rutherford_dsigma_domega, impact_parameter

theta = np.radians([10, 30, 60, 90])
dsigma = rutherford_dsigma_domega(theta, Z1=2, Z2=79, E_kin=5.0)
print(dsigma)
# [5.760e+01 7.406e-01 5.317e-02 1.329e-02]

The cross section at 10 degrees is roughly 4000 times larger than at 90 degrees – almost all alpha particles pass through with only a small deflection, and only a rare few, at small impact parameter, scatter sharply backward. That steep falloff obeys the model’s exact \(1/\sin^4(\theta/2)\) law: multiplying it back out gives the same constant at every angle,

print(dsigma * np.sin(theta / 2) ** 4)
# [0.00332342 0.00332342 0.00332342 0.00332342]  -- constant, as expected

and the corresponding classical impact parameters, impact_parameter(), shrink smoothly as the scattering angle grows – a closer pass means a sharper deflection:

print(impact_parameter(theta, Z1=2, Z2=79, E_kin=5.0))
# [1.318 0.430 0.200 0.115]

Where nuclear stability peaks#

binding_energy_per_nucleon() implements the semi-empirical (Weizsacker) mass formula, whose competing volume, surface, Coulomb, and asymmetry terms combine to produce a broad peak in stability around the iron group:

from physicskit.particle.nuclear import binding_energy_per_nucleon

print(binding_energy_per_nucleon(2, 4))     # He-4:   5.71 MeV/nucleon
print(binding_energy_per_nucleon(26, 56))   # Fe-56:  8.85 MeV/nucleon
print(binding_energy_per_nucleon(82, 208))  # Pb-208: 7.86 MeV/nucleon

Iron-56 sits close to the top of this curve: light nuclei gain binding energy per nucleon by fusing (the volume term dominates, not yet offset by much surface energy), while heavy nuclei lose it by fissioning (the Coulomb repulsion between an ever-larger number of protons eventually wins) – the same competition that makes stellar fusion stop producing net energy once its core has burned up to the iron group.

See Also#