Kinematics, cross-sections, and resonance structure of reactions among atomic nuclei — the bridge from nuclear structure to astrophysical nucleosynthesis and applied nuclear engineering. Core conservation laws: baryon number A, charge Z,…
nuclear-reactions
Q-value from mass defect: Q = (Σm_i − Σm_f)·c²
The Q-value of a nuclear reaction a + X → Y + b + … is the energy released (Q > 0) or required (Q < 0) in the transformation, equal to c²…
Breit-Wigner 1936 resonance: σ(E) Lorentzian with FWHM Γ
Breit-Wigner 1936 (Phys. Rev. 49:519) derived the energy-dependent cross-section for a reaction proceeding through an isolated…
Gamow 1928 Coulomb-barrier tunnelling → stellar-fusion peak
Gamow 1928 (Z. Physik 51:204) applied quantum tunnelling to the Coulomb barrier between two charged nuclei, deriving the penetration…
²²⁶Ra → ²²²Rn + α Q-value: 5.230·10⁻³ u·c² exact (sym. rational)
Closed-form Q-value computation for the α-decay ²²⁶Ra → ²²²Rn + α using tabulated atomic masses (AME2020; exact to 6 significant figures,…
Breit-Wigner FWHM = Γ exact (half-maximum solves to ±Γ/2)
Exact closed-form derivation of the full width at half maximum of the Breit-Wigner Lorentzian line shape. Lorentzian normalised peak:…
Gamow peak E_0 = (E_G·(k_BT)²/4)^{1/3} exact saddle point
Exact closed-form derivation of the stellar-fusion Gamow-peak energy. The integrand of the thermal reaction rate ∝ exp(−E/k_BT) ·…
Gamow-peak saddle-point: maximum of f(E) = -E/kT - b/sqrt(E) at E_0 = (b kT / 2)^(2/3) * 2^(1/3)
Gamow peak framework for stellar nuclear burning (Gamow 1928; Burbidge-Burbidge-Fowler-Hoyle 1957 Rev Mod Phys 29, 547). Setup: the…
Breit-Wigner Lorentzian sum rule: int_{-inf}^{inf} (gamma/2)^2 / ((E-E_R)^2 + (gamma/2)^2) dE = pi gamma / 2
Breit-Wigner resonance framework (Breit-Wigner 1936 Phys Rev 49, 519). Setup: in the neighbourhood of a compound-nucleus resonance at…
Bethe level-density back-shifted Fermi-gas model: S(E) = 2 sqrt(aE); nuclear temperature T = sqrt(E/a)
Bethe back-shifted Fermi-gas level-density framework (Bethe 1936 Phys Rev 50, 332). Setup: for an excited compound nucleus at excitation…
Theorem: Gamow-peak saddle E_0 = 2^(1/3) b^(2/3) (k_BT)^(2/3) / 2 (unique positive real root)
Theorem (Gamow-peak saddle canonical): solving the stationarity condition f'(E_0) = 0 for f(E) = -E/k_BT - b/sqrt(E) gives E_0 = 2^(1/3) *…
Theorem: Lorentzian sum-rule integral = pi * gamma / 2 (residue theorem, contour in upper half-plane)
Theorem (Breit-Wigner integral canonical): for a Lorentzian line-shape (gamma/2)^2 / ((E - E_R)^2 + (gamma/2)^2) with positive width gamma,…
Theorem: Bethe Fermi-gas nuclear temperature T = sqrt(E)/sqrt(a) (thermodynamic 1/T = dS/dE)
Theorem (Bethe nuclear-temperature canonical): the Hardy-Ramanujan / Fermi-gas leading entropy S(E) = 2 sqrt(a E) defines the…
Compound nucleus (Bohr 1936)
N Bohr 1936 compound-nucleus model; statistical-equilibrium before decay; modern Hauser-Feshbach 1952 cross-section formalism.
Optical model (1949)
Fernbach-Serber-Taylor 1949 nucleon-nucleus complex potential; absorption + elastic scattering; modern Koning-Delaroche 2003 global…
Fission (Bohr-Wheeler 1939)
N Bohr-J A Wheeler 1939 liquid-drop-fission theory; modern actinide-fission + super-heavy + neutron-induced fragmentation.
r-process (B2FH 1957)
Burbidge-Burbidge-Fowler-Hoyle 1957 r-process rapid-neutron-capture; site neutron-star-mergers Kasen 2017 GW170817 confirmation.
RIB facilities
GSI + RIKEN-RIBF + GANIL-SPIRAL2 + FRIB-2022 + ATLAS-CARIBU + ISOLDE-CERN; modern neutron-rich nuclei beyond stability +…
Super-heavy elements
Flerov-Oganessian 1960s+ Z=104-118 cold-fusion + hot-fusion; Og-118 last named 2016; predicted island-of-stability around Z=120.
Fermi neutron-induced (1939)
E Fermi 1939 (Nobel 1938) neutron-induced fission + 1942 CP-1 first-reactor; modern reactor-design + thorium + breeder + SMR.
Rutherford transmutation (1919)
E Rutherford 1919 (Nobel 1908) α+N→O+p first nuclear-reaction; modern modern accelerator-driven transmutation + waste-burnup.
Bethe-Weizsäcker (1939)
H Bethe-C von Weizsäcker 1939 (Bethe Nobel 1967) p-p / CNO-cycle stellar-fusion; modern foundational-text in stellar-nucleosynthesis.
Hauser-Feshbach (1952)
W Hauser-H Feshbach 1952 statistical compound-nucleus reaction theory; modern modern TALYS + EMPIRE evaluation-codes nuclear data.
r-process (B²FH 1957)
B²FH Burbidge-Burbidge-Fowler-Hoyle 1957 r-process s-process p-process; modern GW170817-kilonova confirmation + heavy-element origin.
ITER (1985)
Reagan-Gorbachev 1985 + ITER 2007 25k tonnes + 500 MW; modern Q≥10 deuterium-tritium fusion 2025+ first-plasma deferred.