Nuclei and the Making of the Elements

Where do the elements come from? The Physical Nova leaves protons, neutrons and electrons; every element heavier than helium was made later, in stars. The strong force of Part 2.10 binds nuclei, and the stars of Part 2.9 are the furnaces that build them.

The First Three Minutes

While the universe was hotter than about ten billion degrees, protons and neutrons turned freely into one another. As it cooled, the weak force could no longer keep up, and the ratio of neutrons to protons froze, set by the small difference in their masses.

Neutron Freeze-Out G

n⊖ / p⊕ = exp(−Δm · c² / (k_B · T⊣))

Dimensional analysis:
Where (from the book): n⊖ [∅], p⊕ [∅], Δm [𝕄], c [𝕃·𝕋⁻¹], T⊣ [Θ]
[∅] ÷ [∅] = [∅]
exp([𝕄] × [𝕃²·𝕋⁻²] ÷ ([𝕄·𝕃²·𝕋⁻²·Θ⁻¹] × [Θ])) = exp([𝕄·𝕃²·𝕋⁻²] ÷ ([𝕄·𝕃²·𝕋⁻²·Θ⁻¹] × [Θ])) = exp([𝕄·𝕃²·𝕋⁻²] ÷ [𝕄·𝕃²·𝕋⁻²]) = exp([∅]) = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The neutrons left when the weak force stops keeping pace

Where:

• n⊖ [∅] — the number of neutrons
• p⊕ [∅] — the number of protons
• Δm [𝕄] — the neutron–proton mass difference, 1.293 MeV/c²
• c [𝕃·𝕋⁻¹] — light speed
• k_B [𝕄·𝕃²·𝕋⁻²·Θ⁻¹] — the Boltzmann constant
• T⊣ [Θ] — the freeze-out temperature, about 9 × 10⁹ K

➢ About one neutron to six protons at freeze-out, one to seven after some decay; nearly all of those neutrons end in helium, so the gas is a quarter helium by mass (Alpher, Bethe & Gamow, 1948; Cyburt et al., 2016): the Primordial Gas Composition of Part 2.9.

Primordial Helium G

Y⊚ = 2 · x / (1 + x)

Dimensional analysis:
Where (from the book): Y⊚ [∅], x [∅]
[∅]
[∅] × [∅] ÷ ([∅] + [∅]) = [∅] ÷ ([∅] + [∅]) = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

The share of the first gas that is helium, from the neutrons left

Where:

• Y⊚ [∅] — the helium mass fraction, 0.245 as measured
• x [∅] — the neutron-to-proton ratio when nuclei form, about 1/7

➢ Every neutron is bound into helium with a proton, so helium is twice the neutrons' share. That the answer is a quarter, not a tenth or a half, is the 1.293 MeV gap between neutron and proton: change it a little and the universe would be all hydrogen or all helium.

The Binding of Nuclei

Nuclear Binding Energy G

B⊛ = a⊗ · A − a⊡ · A^(2/3) − a⊕ · Z² / A^(1/3) − a⊖ · (A − 2 · Z)² / A

Dimensional analysis:
Where (from the book): B⊛ [𝕄·𝕃²·𝕋⁻²], A [∅], Z [∅], a⊗ [𝕄·𝕃²·𝕋⁻²], a⊡ [𝕄·𝕃²·𝕋⁻²], a⊕ [𝕄·𝕃²·𝕋⁻²], a⊖ [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] × [∅] - [𝕄·𝕃²·𝕋⁻²] × [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅]^([∅]) - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅] - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] ÷ [∅]^([∅] ÷ [∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] ÷ [∅]^([∅]) - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] ÷ [∅] - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅] × [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × ([∅] - [∅])² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅]² ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] × [∅] ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] ÷ [∅] = [𝕄·𝕃²·𝕋⁻²] - [𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²]
[𝕄·𝕃²·𝕋⁻²] = [𝕄·𝕃²·𝕋⁻²] ✓ Valid PulseCore Verified ✓

How tightly a nucleus is bound: a liquid drop of nucleons

Where:

• B⊛ [𝕄·𝕃²·𝕋⁻²] — the binding energy of a nucleus
• A [∅] — its number of nucleons
• Z [∅] — its number of protons
• a⊗ [𝕄·𝕃²·𝕋⁻²] — the volume term, about 15.8 MeV
• a⊡ [𝕄·𝕃²·𝕋⁻²] — the surface term, about 18.3 MeV
• a⊕ [𝕄·𝕃²·𝕋⁻²] — the electric term, about 0.71 MeV
• a⊖ [𝕄·𝕃²·𝕋⁻²] — the asymmetry term, about 23.2 MeV

➢ Weizsäcker's formula (1935): the strong force binds each nucleon to its neighbours, the surface binds less, and the protons' charge pushes apart. Binding per nucleon rises to a peak at iron and nickel and falls after: lighter nuclei release energy by fusing, heavier ones by splitting. That peak is why stars stop at iron.

The Furnaces

Stars build the elements in stages (Burbidge, Burbidge, Fowler & Hoyle, 1957). Hydrogen fuses to helium. Helium fuses to carbon only through a narrow door: three helium nuclei meet through a resonance in carbon at 7.65 MeV, the Hoyle state, predicted by Hoyle (1954) from the fact that carbon exists and then found; without it there would be almost no carbon or oxygen. Carbon and helium build oxygen, neon, magnesium, silicon, and the chain ends at iron in the cores of massive stars, whose collapse is the supernova of Part 2.9. Heavier elements are built by neutron capture: slowly, in giant stars, and rapidly, in the mergers of neutron stars, where gold and platinum are made (Abbott et al., 2017). Lithium, beryllium and boron are mostly chipped from heavier nuclei by cosmic rays.

Where It Meets the Pulse

The elements are made by two forces and the stars: the strong force sets every nucleus and, through Part 2.10, follows the universe's level; the weak force sets the first gas; the stars, sized by the same ratio of Pulse to matter (Part 2.9), do the rest. A universe's chemistry is therefore written by its level long before its first atom forms.