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29 formulas

Nuclear & Particle

Decay, fission, quarks. Every formula below opens into a live, hands-on simulation.

relativistic energy
E=mc2E = mc^2

Mass–Energy Equivalence

Mass is a highly concentrated form of energy; converting even a tiny mass releases enormous energy.

radioactivity
N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

Radioactive Decay Law

Undecayed nuclei fall exponentially; each has a fixed decay probability per unit time.

radioactivity
T1/2=ln⁡2λT_{1/2} = \frac{\ln 2}{\lambda}

Half-Life Relation

Half-life is a fixed isotope property—independent of the sample size.

nuclear structure
B=Δm c2B = \Delta m \, c^2

Nuclear Binding Energy

The missing mass when nucleons bind into a nucleus appears as the energy holding them together.

nuclear reactions
Q=(mi−mf)c2Q = (m_i - m_f) c^2

Q-Value of Nuclear Reaction

Lighter products mean the missing mass exits as kinetic energy and radiation.

relativistic energy
E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

Relativistic Energy–Momentum Relation

Energy and momentum combine to form a Lorentz invariant — the rest mass of the particle.

particle physics
λ=hp\lambda = \frac{h}{p}

de Broglie Wavelength (Relativistic)

A particle’s wavelength shrinks with momentum—more momentum probes smaller scales.

radioactivity
log⁡10T1/2=aZQ+b\log_{10} T_{1/2} = a \frac{Z}{\sqrt{Q}} + b

Geiger–Nuttall Law

Alpha half-lives vary exponentially with decay energy via quantum tunneling.

nuclear structure
B(A,Z)=aVA−aSA2/3−aCZ2A1/3−aA(A−2Z)2A+δ(A,Z)B(A,Z) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta(A,Z)

Semi-Empirical Mass Formula

Five competing terms—volume, surface, Coulomb, asymmetry, pairing—model the nuclear binding energy.

particle physics
dσdΩ=12α2rc2(ω′ω)2[ωω′+ω′ω−sin⁡2θ]\frac{d\sigma}{d\Omega} = \frac{1}{2} \alpha^2 r_c^2 \left(\frac{\omega'}{\omega}\right)^2 \left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right]

Klein–Nishina Cross Section

QED correction to Thomson scattering: photon cross-section shrinks at high energy.

photon interactions
Δλ=hmec(1−cos⁡θ)\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Compton Scattering Wavelength Shift

A photon hitting an electron transfers momentum, so it leaves with a longer wavelength — light behaves like a particle.

atomic structure
a0=4πϵ0ℏ2mee2a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2}

Bohr Radius

The natural size of an atom emerges from balancing electron kinetic energy with Coulomb attraction to the nucleus.

atomic structure
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Hydrogen emits sharp, discrete colors because its electron only inhabits quantized energy levels — the spectrum is the atom's barcode.

photon interactions
Kmax=hf−ϕK_{max} = h f - \phi

Photoelectric Effect Equation

Light comes in packets of energy hf; only packets above the work-function threshold can free an electron — no matter how intense the beam.

radioactivity
A=λNA = \lambda N

Radioactive Activity-Decay Constant Relation

Activity is just the number of unstable atoms times how likely each is to decay per second.

scattering
dσdΩ=(Z1Z2e216πϵ0E)21sin⁡4(θ/2)\frac{d\sigma}{d\Omega} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)}

Rutherford Scattering Cross Section

Rare large-angle scattering proves the atom has a tiny, dense, positively charged core — the nucleus.

nuclear force
V(r)=−g2e−mrc/ℏrV(r) = -g^2 \frac{e^{-m r c / \hbar}}{r}

Yukawa Potential

A short-ranged attractive force whose range is set by the mass of the exchanged particle — heavier mediators = shorter range.

transition rates
Γi→f=2πℏ∣Mfi∣2ρ(Ef)\Gamma_{i \to f} = \frac{2\pi}{\hbar} |M_{fi}|^2 \rho(E_f)

Fermi's Golden Rule

The transition rate between quantum states is proportional to the squared coupling times the number of final states available.

photon interactions
Eγmin⁡=2mec2E_{\gamma}^{\min} = 2 m_e c^2

Pair Production Threshold Energy

A photon must carry at least the combined rest energy of the electron and positron it creates — 1.022 MeV.

radioactive decay
Qα=(mP−mD−mα)c2Q_\alpha = (m_P - m_D - m_\alpha)c^2

Alpha Decay Q-Value

The kinetic energy carried away by the alpha particle (and the recoiling daughter) equals the rest-mass difference between parent and decay products, expressed via E = mc².

weak interaction
dNdE=GF2∣M∣22π3ℏ7c5F(Z,E)pE(Q−E)2\frac{dN}{dE} = \frac{G_F^2 |M|^2}{2\pi^3 \hbar^7 c^5} F(Z,E) p E (Q - E)^2

Fermi Beta Decay Spectrum

Three-body decay (nucleus + electron + neutrino) shares the released energy Q, so the electron's spectrum is continuous; the (Q-E)² factor is the neutrino phase space.

scattering
λ=1nσ\lambda = \frac{1}{n\sigma}

Mean Free Path

If the medium has n targets per unit volume each presenting cross-sectional area σ, the average distance between interactions is 1/(nσ).

nuclear structure
Bˉ=B(A,Z)A\bar{B} = \frac{B(A,Z)}{A}

Binding Energy per Nucleon

Dividing total binding energy by mass number gives the per-nucleon glue strength; the curve peaks near A=56 (Fe/Ni), explaining why fusion liberates energy below A=56 and fission above.

radioactive decay
τ=1λ\tau = \frac{1}{\lambda}

Mean Lifetime of Radioactive Nuclei

Mean lifetime is the expectation value of the exponential decay distribution; equivalently, the time after which N drops to N₀/e.

nuclear structure
R=R0A1/3R = R_0 A^{1/3}

Nuclear Radius Formula

Nuclei have nearly constant density, so volume scales linearly with A and radius scales as A^(1/3) — like balls of incompressible nuclear fluid.

scattering
σT=8π3(e24πε0mec2)2\sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e c^2}\right)^2

Thomson Scattering Cross Section

An EM wave shakes a free electron, which re-radiates a dipole pattern; the total scattering rate corresponds to an effective area σ_T set by the classical electron radius r_e.

radiation
v>cn  ⇔  cos⁡θC=1nβv > \frac{c}{n} \;\Leftrightarrow\; \cos\theta_C = \frac{1}{n\beta}

Cherenkov Radiation Threshold

When a charged particle moves faster than the phase velocity of light in the medium (c/n), it sets off a coherent optical shockwave at angle θ_C — the blue glow in reactor pools.

particle physics
p=uud,n=udd,ZAX:  A=Z+Np = uud, \quad n = udd, \quad {}^{A}_{Z}X: \; A = Z + N

Building Atoms from the Standard Model

Everything you've ever touched is three particles: up quarks, down quarks, and electrons. Two ups and a down make a proton (+1); one up and two downs make a neutron (0). The proton count Z picks the element, the neutron count N picks the isotope, and the electron count sets the charge. Stray too far from the stable Z–N balance and the weak force flips a quark — beta decay — to restore it.

fission
Nn=N0 kn,k=ν pN_{n} = N_{0}\,k^{n}, \qquad k = \nu \, p

Nuclear Chain Reaction

One neutron splits a uranium-235 nucleus, releasing ~200 MeV and ν ≈ 2.4 fresh neutrons. If, on average, more than one of those goes on to cause another fission (k > 1), the population explodes geometrically — 2, 4, 8, … 2⁸⁰ in microseconds. A reactor is the art of pinning k at exactly 1.000; a bomb is k ≈ 2 with nothing in the way.