ZeptoPhysics: The Precision Frontier
Measuring the smallest effects in nature
A zeptometer is 10⁻²¹ m: about a millionth of a proton's radius. No experiment images anything that small. Instead, distance and energy are two sides of one coin (λ = ħc/E), so 1 zm corresponds to roughly 200 TeV, about fifteen times the LHC's collision energy. Physicists reach that scale indirectly, by measuring rare decays, magnetic moments and symmetry violations so precisely that the faint influence of much heavier, undiscovered particles would show up as a deviation.
The same mindset drives the rest of this site: gravitational-wave detectors resolve strains of 10⁻²¹, pulsars are timed to below a microsecond, and plasma and collider instruments push measurement limits. Every toolkit here lets you compute those numbers yourself.
Heavy particles can appear briefly as "virtual" contributions inside rare decays, nudging rates and angles away from Standard Model predictions.
Do electrons, muons and taus behave identically? Most tests now agree with the Standard Model; B-meson decays to taus remain under scrutiny.
Dark matter, the matter–antimatter imbalance and neutrino masses all point beyond the Standard Model.
Pulsars, gravitational waves, plasmas, colliders, extreme lasers, fusion, atomic clocks and frontier machines: every number here can be recomputed in your browser.
Energy ↔ distance ↔ which experiment can reach it
λ = ħc / E with ħc = 197.327 MeV·fm. Enter a mass scale for a hypothetical particle or a length, and see where it sits on the ladder.
Indirect reaches are order-of-magnitude figures assuming generic O(1) couplings; weaker or flavour-aligned couplings lower them a lot. Direct LHC reach for new particles is a few TeV, well below √s.
The questions driving precision physics
Where to see what the big experiments are doing right now
Each lab publishes its own live status. These links go straight to the official pages.
Proton collisions at 13.6 TeV (Run 3), bunch crossings every 25 ns. LHC Page 1 shows beam mode, energy and luminosity.
LHC Page 1 ↗ LIGO · Virgo · KAGRAKilometre-scale interferometers resolving strains near 10⁻²¹. GraceDB lists public gravitational-wave candidates as they arrive.
Public alerts (GraceDB) ↗ FermilabThe 120 GeV Main Injector drives neutrino beams (NuMI, future LBNF/DUNE) and muon experiments. The accelerator notice board shows beam status.
Accelerator status ↗ IceCube & multi-messengerA cubic-kilometre neutrino detector in the South Pole ice. Its realtime alerts, with gamma-ray and GW alerts, flow through NASA's GCN.
GCN alerts ↗Recurring physics & astrophysics conferences
Key experiments probing physics beyond the Standard Model
Rare B-meson decays, CP violation and the ultra-rare K⁺ → π⁺νν̄ decay, the cleanest probes of quantum-loop effects.
Muon g-2 published its final measurement in 2025 (~127 ppb). Mu2e, a search for muon-to-electron conversion, is being commissioned.
An electron–positron B factory testing tau-vs-muon universality (R(D), R(D*)) and searching for dark-sector particles.
Searching for μ → eγ, forbidden in the Standard Model, using the world's most intense continuous muon beam.
A multi-tonne liquid-xenon detector a mile underground, looking for dark-matter particles scattering off nuclei.
A global interferometer network turning 10⁻²¹ strains into black-hole and neutron-star observations.
ZeptoPulse Computational Bench
Magnetic Dipole Spin-Down Solver
How it works: Magnetic dipole spin-down
- τc = P / (2Ṗ)
- Bs ≈ 3.2×10¹⁹ √(PṖ) G
- Ė = 4π² I Ṗ / P³, I = 10⁴⁵ g cm²
- RLC = cP / 2π
- Assumes pure magnetic-dipole braking (braking index n = 3) and a birth period much shorter than today’s.
- Bs is the equatorial surface field for a 10 km neutron star.
- Characteristic age is an estimate, not a true age.
Worked example · Crab pulsar, P = 33.3 ms, Ṗ = 4.21×10⁻¹³: τc ≈ 1,250 yr, but the supernova was seen in 1054 AD (≈ 970 yr ago). The gap shows the n = 3 and short-birth-period assumptions are only approximate.
References: ATNF Pulsar Catalogue · Lorimer & Kramer (2005), Handbook of Pulsar Astronomy
Relativistic FLRW Cosmology Calculator (H₀ = 67.4, Ω_m = 0.315)
How it works: Flat ΛCDM distances
- E(z) = √(Ωm(1+z)³ + ΩΛ)
- DC = (c/H₀) ∫₀z dz′/E(z′)
- DL = (1+z) DC
- tL = (1/H₀) ∫₀z dz′/[(1+z′)E(z′)]
- Planck 2018 parameters: H₀ = 67.4 km/s/Mpc, Ωm = 0.315, flat universe.
- Radiation neglected (error well under 1% for z < 10).
- β is the special-relativistic Doppler equivalent of z, shown for intuition. Cosmological redshift is really due to expansion, not motion.
Worked example · At z = 1: DC ≈ 3,400 Mpc, DL ≈ 6,800 Mpc, and the light left its source about 7.9 billion years ago.
References: Hogg, Distance measures in cosmology · Planck 2018 results VI
Schwarzschild Horizon Solver (non-spinning)
How it works: Schwarzschild horizon scales
- Rs = 2GM/c² ≈ 2.95 km × (M/M☉)
- Rph = 1.5 Rs, RISCO = 3 Rs
- TH ≈ 6.17×10⁻⁸ K × (M☉/M)
- Non-spinning black hole. For a maximally spinning (Kerr) hole the prograde ISCO shrinks to Rs/2.
- Hawking temperature for stellar black holes is far colder than the 2.7 K microwave background, so they absorb more than they radiate.
Worked example · Sgr A* (≈ 4.3 million M☉): Rs ≈ 1.3×10⁷ km, about 0.08 AU, roughly a fifth of Mercury’s orbit.
References: Event Horizon Telescope · Misner, Thorne & Wheeler, Gravitation
Fast Radio Burst (FRB) Cold Plasma Dispersion Solver
How it works: Cold-plasma dispersion
- Δt ≈ 4.149×10³ s × DM × (νlo⁻² − νhi⁻²), ν in MHz
- DM = ∫ ne dl (pc cm⁻³)
- z ≈ (DM − DMMW+host) / ~900 pc cm⁻³
- Delay formula is exact for a cold, unmagnetised plasma.
- The redshift estimate uses the Macquart relation with a fixed 60 pc cm⁻³ for the Milky Way and host. Real scatter is large (±30% or more).
Worked example · DM = 560 pc cm⁻³ across 400–800 MHz: the low end arrives ≈ 10.9 s after the high end.
References: Macquart et al. 2020, Nature · Lorimer & Kramer (2005), ch. 4
ZeptoWaves Spacetime Bench
Binary Inspiral Chirp Mass & Strain Amplitude Solver
How it works: Leading-order binary inspiral
- 𝓜 = (m₁m₂)3/5 / (m₁+m₂)1/5
- h₀ = 4 (G𝓜/c²)5/3 (π f / c)2/3 / D
- τ ≈ 2.18 s (1.21 M☉/𝓜)5/3 (100 Hz/f)8/3
- fISCO ≈ 4,400 Hz × (M☉/M), ΔL = h L / 2
- LGW = (32/5)(c⁵/G)(πG𝓜f/c³)10/3
- Newtonian (quadrupole) order: good early in the inspiral, increasingly off near merger.
- h₀ is for an optimally oriented source directly overhead; average sources are ~2.5× weaker.
- f is the gravitational-wave frequency (twice the orbital frequency).
- Distance is luminosity distance; for distant sources the detector sees a redshifted chirp mass.
Worked example · GW150914 (36 + 29 M☉ at 410 Mpc, f = 150 Hz): 𝓜 ≈ 28 M☉, h₀ ≈ 2×10⁻²¹ for ideal orientation (observed peak ≈ 1×10⁻²¹), and ΔL ≈ 4×10⁻¹⁸ m over a 4 km arm.
References: Abbott et al. 2016, PRL 116, 061102 · GWOSC open data
ZeptoPlasma Laboratory
Relativistic Synchrotron & Debye Shielding Solver
How it works: Synchrotron & plasma scales (CGS)
- νc = (3/2) γ² νg, νg = 2.8 MHz × B[G]
- tcool ≈ 5.1×10⁸ / (γ B²) s
- λD ≈ 6.9 √(T/ne) cm, fpe ≈ 8.98 kHz √ne
- rL ≈ 1.7×10³ γ / B cm, beaming ≈ 1/γ
- Electrons with 90° pitch angle; an isotropic distribution cools ~1.5× slower.
- Synchrotron losses only (no inverse-Compton or adiabatic cooling).
- T in kelvin, ne in cm⁻³, B in gauss.
Worked example · γ = 1000 in a 1.5 G field: νc ≈ 6.3 GHz (radio) and the electron radiates away its energy in about 2.3×10⁵ s (≈ 2.6 days).
References: NRL Plasma Formulary · Rybicki & Lightman (1979), Radiative Processes in Astrophysics
Zeptomic Subatomic Bench
Relativistic Collider Kinematics & Magnetic Spectrometer
Standard Model Boson Decay Lifetimes (τ = ħ/Γ)
How it works: Collider kinematics
- √s = 2Ebeam (symmetric head-on beams)
- γ = Ebeam / mpc²
- R[m] = pT[GeV/c] / (0.3 B[T]) (unit charge)
- τ = ħ / Γ, ħ = 6.582×10⁻²⁵ GeV s
- √s is the proton–proton energy; individual quark and gluon collisions carry only a fraction of it.
- Widths are PDG values; the Higgs width shown is the Standard Model prediction (~4 MeV), not a direct measurement.
Worked example · LHC Run 3 (6.8 TeV per beam): √s = 13.6 TeV and γ ≈ 7,250. A 45 GeV muon in the 3.8 T CMS solenoid curls with a radius of ≈ 39 m.
References: Particle Data Group · CERN: LHC facts
ZeptoLaser: Extreme Light Bench
Squeeze a few joules into a few femtoseconds and a few microns and the light field becomes strong enough to make electrons relativistic, accelerate them in centimetres, or create attosecond flashes. These calculators follow one laser pulse through each regime.
Focused pulse
Gaussian pulse in time and space; linear polarisation
Laser wakefield accelerator
Bubble-regime scalings (Lu et al. 2007) driven by the pulse above
Strong-field QED probe
Electron beam colliding head-on with the focused pulse
High-harmonic generation & attosecond pulses
Gas-phase HHG at moderate intensity (three-step model)
How it works: Extreme-light scalings
- P ≈ 0.94 E / τ, I₀ ≈ 0.88 P / d² (Gaussian, FWHM)
- a₀ = 0.855 λ[µm] √(I / 10¹⁸ W cm⁻²)
- E_L = 2.75 kV/m × √(I [W/cm²]), E_S = 1.32×10¹⁸ V/m
- n_c = 1.1×10²¹ / λ²[µm] cm⁻³, λp = 33 µm × (10¹⁸/n)^{1/2}
- ΔW ≈ 1.7 GeV (P/100 TW)^{1/3} (10¹⁸/n)^{2/3} (0.8/λ)^{4/3}
- χ ≈ 2γ E_L / E_S (head-on)
- E_cut = Iₚ + 3.17 Uₚ, Uₚ = 9.33×10⁻¹⁴ I λ² eV, Δt·ΔE ≥ 1.82 eV·fs
- Ideal Gaussian beams; real focal spots put only ~30–50% of the energy in the central peak.
- Wakefield energy gain assumes a matched, guided pulse with P > P_c over the dephasing length; real accelerators need guiding over centimetres.
- χ ≈ 1 marks the onset of quantum radiation reaction and pair production by colliding beams.
- The attosecond figure is the transform limit of the whole plateau; isolated pulses also need gating, and the shortest measured are a few tens of attoseconds.
Worked example · A 5 J, 30 fs Ti:sapphire pulse focused to 3 µm reaches ≈ 1.5×10²¹ W/cm² (a₀ ≈ 27). In a 10¹⁸ cm⁻³ plasma it can drive electrons to ≈ 2 GeV over a couple of centimetres, an energy that takes roughly 40 m of conventional RF linac.
References: Esarey, Schroeder & Leemans, Rev. Mod. Phys. 81, 1229 (2009) · Lu et al., PRST-AB 10, 061301 (2007) · Nobel Prize in Physics 2023 (attosecond pulses)
ZeptoFusion: Burning Plasma Bench
Fusion needs a plasma hot enough for nuclei to overcome their repulsion, dense enough to collide often, and confined long enough to pay back the heating. These calculators turn those three numbers into fusion power and gain.
Magnetic confinement (tokamak)
Zero-dimensional power balance, flat profiles, D-T fuel
Inertial confinement (laser driven)
Target gain vs. what a power plant needs
How it works: D-T power balance
- P_fus = ¼ (f n)² ⟨σv⟩ × 17.6 MeV
- ⟨σv⟩ from the Bosch–Hale (1992) fit
- W = 3 n T, P_ext = W/τ_E − P_α, P_α = P_fus/5
- Lawson: n T τ_E ≳ 3×10²¹ keV s m⁻³ for ignition
- β = 2μ₀ (2 n T) / B², P_br = 5.35×10⁻³⁷ Z_eff n² √T W m⁻³
- Flat (volume-averaged) profiles. Real peaked profiles give more fusion power for the same averages, which matters most for compact high-field designs like SPARC: this model gives Q ≈ 2 for its average parameters, while its peaked-profile design projects Q ≈ 10.
- τ_E is the measured or design confinement time, which already includes radiation losses, so bremsstrahlung is shown for reference only.
- Fuel fraction accounts for impurities and helium ash diluting the D-T.
- ICF engineering gain uses laser wall-plug efficiency only; a plant also needs tritium breeding and thermal conversion (~40%).
Worked example · ITER design values (n = 10²⁰ m⁻³, T = 8.9 keV, τ_E = 3.7 s) give nTτ_E ≈ 3.3×10²¹ and Q ≈ 9, close to ITER’s Q = 10 goal. NIF’s December 2022 shot made 3.15 MJ from 2.05 MJ of laser light (G ≈ 1.5), but its lasers drew roughly 300 MJ from the grid.
References: Bosch & Hale, Nucl. Fusion 32, 611 (1992) · ITER · LLNL National Ignition Facility
Frontier Technology Bench
The machines behind the next discoveries: free-electron lasers that film atoms, high-temperature superconducting magnets that shrink fusion reactors and colliders, and the colliders being designed to follow the LHC.
X-ray free-electron laser
Undulator resonance for a relativistic electron beam
High-field superconducting magnet
Forces and stored energy in a solenoid bore
Circular collider designer
Beam energy, ring size and the two killers: synchrotron radiation and decay
How it works: Undulators, magnets & rings
- λ_x = (λᵤ / 2γ²)(1 + K²/2), E_ph = 1.2398 keV·nm / λ
- p_mag = B² / 2μ₀, E = (B²/2μ₀) π r² L
- p [GeV/c] = 0.3 B[T] ρ[m], C = 2πρ / fill
- U₀ = 88.5 keV × E⁴[GeV] / ρ[m] × (m_e/m)⁴
- τ_lab = γ τ₀, τ₀(μ) = 2.197 µs
- On-axis fundamental only; real XFEL output also depends on beam emittance, energy spread and peak current.
- Magnet stored energy assumes a uniform field inside a long solenoid; real coils store extra energy in the fringe field.
- Fusion power density at fixed β scales as B⁴, which is why high-temperature superconductors (REBCO) changed compact-tokamak designs.
- Fill factor is the fraction of the ring occupied by dipoles (LHC ≈ 0.66).
Worked example · At 6.8 TeV in 8.33 T dipoles the LHC needs a 2.7 km bending radius and a 26.7 km ring. An electron at the same energy would lose far more than its own energy to synchrotron radiation every turn, which is why e⁺e⁻ machines stop near 100–200 GeV and why a muon (207× heavier) collider is attractive, if the muons can be used within their ~2,000-turn lifetime.
References: LCLS at SLAC · European XFEL · FCC study · International Muon Collider Collaboration · NHMFL
ZeptoClock: Time as a Precision Instrument
The best atomic clocks now agree to about one part in 10¹⁸: they would drift by less than a second over twice the age of the universe. At that level a clock feels a 1 cm change in height, and pairs of clocks become detectors for changing constants of nature and some kinds of dark matter.
Gravity clock (gravitational redshift)
A clock higher in Earth’s gravity runs faster
Satellite clocks & GPS
Gravity speeds orbiting clocks up, orbital speed slows them down
Clock stability & averaging time
Precision improves as 1/√τ until it hits the clock’s systematic floor
Are the constants changing? Nuclear vs. atomic clocks
Each clock’s frequency responds differently to a change in the fine-structure constant α
Ultralight dark matter
Wave-like dark matter would make "constants" oscillate at its Compton frequency
Atom interferometer
Free-falling atoms as test masses: phase φ = keff g T²
How it works: Clocks as sensors
- Δf/f = ΔU / c² ≈ g Δh / c² = 1.09×10⁻¹⁶ per metre
- Orbit: ΔU term GM(1/R⊕ − 1/r)/c², velocity term −v²/2c², v = √(GM/r)
- σ_y(τ) = σ₁ / √τ (white frequency noise)
- Δf/f = K · Δα/α
- f_DM = m c² / h = 2.42×10¹⁴ Hz × m[eV], τ_coh ≈ 10⁶ / f
- φ = k_eff g T², δφ = 1/√N (shot noise)
- Weak-field gravity; the height formula uses the exact potential, so it stays correct for large heights.
- GPS figures ignore Earth’s rotation and orbit eccentricity (both corrected in the real system).
- Clock instabilities and floors are representative published orders of magnitude; the thorium clock figures are projections.
- Sensitivity coefficients K are theory values; for thorium-229, estimates range from ~10³ to ~10⁴.
- Dark-matter coherence assumes a galactic velocity spread of ~10⁻³ c.
Worked example · Two optical clocks 33 cm apart differ by 3.6×10⁻¹⁷, which NIST resolved in 2010 (Chou et al.). GPS satellite clocks run 38.5 µs/day fast; uncorrected, that is about 11 km of range error per day.
References: Chou et al., Science 329, 1630 (2010) · Ashby, Relativity in the Global Positioning System · Zhang et al., Nature 633, 63 (2024): Th-229 nuclear clock transition · BIPM: redefinition of the second