Nuclear & Particle Physics — formula sheet
Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Semi-Empirical Mass Formula from the Liquid-Drop Model
The binding energy is a competition between a bulk volume gain \(a_V A\) that wants nuclei large, and three penalties: the surface term (fewer bonds at the edge), the Coulomb term (proton–proton repulsion, growing like \(Z^2/A^{1/3}\)), and the asymmetry term (Pauli cost of \(N\neq Z\)). The pairing term is a small even–odd correction. Their balance produces the peak of \(B/A\approx 8.8\) MeV near iron and the drift of the stability line toward neutron excess in heavy nuclei. A representative fit (in MeV): \(a_V\approx 15.8\), \(a_S\approx 18.3\), \(a_C\approx 0.71\), \(a_A\approx 23.2\), \(a_P\approx 12\).
Coulomb Energy of a Uniformly Charged Nucleus
The electrostatic self-energy of a uniformly charged sphere is \(3/5\) of the energy \(Q^2/(4\pi\varepsilon_0 R)\) you would (naively) assign to charge \(Q\) at separation \(R\). The factor \(3/5\) reflects that most charge sits at radii smaller than \(R\), so the effective mutual separation is less than \(R\), raising the energy above that of a single thin shell (which would give a factor \(1/2\)). Substituting \(R = r_0 A^{1/3}\) reproduces the mass-formula scaling \(a_C\, Z^2 A^{-1/3}\), with \(a_C = \tfrac{3}{5}\,e^2/(4\pi\varepsilon_0 r_0) \approx 0.72\text{–}0.77\ \text{MeV}\) once the point-charge correction \(Z^2 \to Z(Z-1)\) and diffuseness are folded in empirically.
Asymmetry Term from the Fermi-Gas Model
Making the neutron and proton Fermi seas unequal — deepening one, shallowing the other at fixed total number — always raises the kinetic energy, and the penalty is quadratic in the imbalance \(N-Z\) and inversely proportional to \(A\) (a larger system dilutes the cost per unit imbalance). The scale is set by the Fermi energy: a third of it. No force enters; this is the Pauli exclusion tax on isospin asymmetry.
The Valley of Beta Stability
At fixed \(A\) the isobars lie on a parabola in \(Z\); the vertex \(Z_*\) is the most stable charge and the valley of stability is the curve \(Z_*(A)\). Because \(a_C/2a_A>0\), the denominator exceeds \(2\), so \(Z_*<A/2\): heavier nuclei are pushed to a neutron excess by the Coulomb energy, which the asymmetry term can only partly resist. Any isobar with \(Z<Z_*\) lowers its mass by \(\beta^-\) decay (\(n\to p\), \(Z\to Z+1\)); any with \(Z>Z_*\) does so by \(\beta^+\)/electron capture (\(Z\to Z-1\)).
Gamow Factor and the Geiger-Nuttall Law
Fermi Theory of Beta Decay and the Kurie Plot
The electron spectrum is a product of three energy-dependent pieces: the outgoing-electron momentum-space weight \( p_e E_e \), the neutrino phase-space tail \( (E_0 - E_e)^2 \), and the Coulomb factor \( F(Z,E_e) \) that pulls electrons toward low energy (\( \beta^- \), attractive daughter) or pushes them up (\( \beta^+ \), repulsive). Everything nuclear collapses into the single constant \( \overline{|M_{\text{nucl}}|^2} \). The Kurie transform strips the kinematics so a straight line falls to zero at \( E_0 \); the total rate scales as the fifth power of the energy release, so a decay with twice the \( Q \)-value is roughly thirty times faster.
Radioactive Decay and Bateman Equations
A single species falls off as a pure exponential with mean life \(1/\lambda\). Any daughter in a chain is a signed superposition of the exponentials of every ancestor and of itself: each mode \(e^{-\lambda_i t}\) is weighted by the total upstream production rate divided by how far that mode's rate sits from all the others. Fast\-decaying intermediates contribute short transients; the slowest ancestor dominates the long\-time tail, so a short\-lived daughter ends up "tracking" a long\-lived parent.
Rutherford Scattering Cross-Section
The number of particles scattered into unit solid angle at angle \(\theta\) falls off as \(\sin^{-4}(\theta/2)\): it is enormous in the forward direction and drops steeply toward back-scatter, yet never vanishes — even \(\theta=\pi\) has a finite, non-zero rate set entirely by \(\kappa/4E\). The cross-section scales as \((z_1 z_2)^2\) (charge product squared) and as \(E^{-2}\) (faster projectiles are deflected less). Crucially, \(\hbar\) is absent: the classical orbit and the quantum Born result are numerically identical for the \(1/r\) potential.
Born Approximation and the Form Factor
Scattering at momentum transfer \(\mathbf q\) reads off a single Fourier component of the potential. A pointlike target scatters with a smooth Rutherford-like amplitude; any spatial extent multiplies it by the form factor \(F(\mathbf q)\), which starts at unity and falls off over \(q\sim1/(\text{size})\). Diffraction minima in \(|F|^{2}\) are the interference of waves scattered from different parts of the distribution. The small-\(q\) slope, \(F\simeq1-\tfrac16 q^{2}\langle r^{2}\rangle\), gives the mean-square radius without any shape assumption.
Mandelstam Invariants and Relativistic Two-Body Kinematics
The Mandelstam variables package all frame-independent kinematics of a \(2\to2\) process into two independent numbers (a third is fixed by the mass sum rule). \(\sqrt{s}\) is the energy available in the CM to make new particles; the threshold formula converts that requirement into a minimum beam energy for a stationary target; and \(p^*\) gives the common magnitude of the back-to-back CM momenta, vanishing exactly as \(\sqrt{s}\to(m_a+m_b)\).
Lorentz-Invariant Phase Space and the Decay/Cross-Section Master Formulae
Klein-Gordon Equation and the Yukawa Potential
A massive mediator produces a Coulomb‑like \(1/r\) potential multiplied by an exponential screening factor \(e^{-\mu r}\). The field is effectively "used up" beyond one screening length \(R=\hbar/mc\), the mediator's reduced Compton wavelength. The heavier the exchanged quantum, the shorter the reach of the force; a massless mediator (\(\mu\to 0\)) restores the infinite‑range Coulomb law. The static potential energy between two such charges is \(V(r) = -\dfrac{g^2}{4\pi}\dfrac{e^{-\mu r}}{r}\) (attractive for like scalar charges), the Fourier transform at zero energy transfer of the scalar propagator \((k^2+\mu^2)^{-1}\).
Dirac Equation, Antiparticles and the g=2 Prediction
Linearising the relativistic energy relation cannot be done with numbers; it requires four anticommuting \(4\times4\) matrices. The resulting four-component wavefunction automatically carries spin \( \tfrac12 \) (two states) and comes in two energy signs (particle and antiparticle, two more). Coupling to a magnetic field produces a magnetic moment exactly twice the naive classical value, \( g=2 \) — a parameter-free prediction. The negative-energy branch, reinterpreted, is the positron.
Noether's Theorem for Conserved Currents
Each continuous symmetry contributes two terms to its current: an internal piece \( \pi^\mu_a\Psi_{r,a} \) built from the canonical momentum \( \pi^\mu_a=\partial\mathcal{L}/\partial(\partial_\mu\phi_a) \) times how the fields change, and a spacetime piece \( -T^\mu{}_\nu X_r^\nu \) built from the stress tensor times how the coordinates move. Pure internal symmetries (\( X=0 \)) give charge/flavour currents; pure translations (\( \Psi=0,\ X_r^\nu=\delta_r^\nu \)) give \( J^\mu{}_\nu=-T^\mu{}_\nu \), i.e. energy–momentum conservation \( \partial_\mu T^\mu{}_\nu=0 \).
Local U(1) Gauge Invariance Fixes the QED Coupling
Demanding that the free Dirac theory be invariant under a spacetime-dependent phase forces three things at once: the derivative must be covariantized, \(\partial_\mu\to D_\mu=\partial_\mu+ieA_\mu\); a gauge field \(A_\mu\) must exist and transform as \(A_\mu\to A_\mu-\tfrac1e\partial_\mu\alpha\); and the matter current \(J^\mu=\bar\psi\gamma^\mu\psi\) must couple to it as \(-eA_\mu J^\mu\). The photon and its coupling to charge are consequences of the symmetry, not inputs. Only the number \(e\) is left free.
Isospin, Hypercharge and the Gell-Mann-Nishijima Relation
Electric charge is not an independent label of a hadron: it is the third isospin component shifted by half the hypercharge. Within one isospin multiplet, \(Y\) is fixed and \(Q\) climbs in unit steps as \(I_3\) runs from \(-I\) to \(+I\); the whole ladder is centred on \(\langle Q\rangle=Y/2\). Different multiplets are distinguished by their centre \(Y/2\), and only \(I_3\) moves you within one.
Spontaneous Symmetry Breaking and the Higgs Mechanism
A spontaneously broken local \(U(1)\) gives the gauge boson a mass equal to the gauge coupling times the vacuum expectation value. The Goldstone mode is not a physical particle here — it is eaten to provide the third (longitudinal) polarization the massive vector needs. What remains physical is one massive scalar \(h\) whose mass is set by the self-coupling \(\lambda\) and the same \(v\).
Nuclear Binding Energy and the Energetics of Fission and Fusion
The binding energy is the mass defect converted to energy; the \(Q\)-value of any reaction is the increase in total binding energy. Fission of heavy nuclei and fusion of light nuclei both move the system toward the peak of the binding-energy-per-nucleon curve (near \(A \approx 56\), \(B/A \approx 8.8\) MeV), releasing energy. Fusion of D–T releases 17.6 MeV per reaction; fission of \(^{235}\)U releases about 200 MeV per fission.