Cyclotron Frequency Calculator Charged Particle
Calculate cyclotron frequency f_c=qB/(2π·m), angular velocity ω_c and Larmor radius r_L. Larmor 1897, CERN, ITER fusion.
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Cyclotron frequency charged particle calculator
The cyclotron frequency calculator determines the rotation frequency f_c of a charged particle (electron, proton, ion) in uniform magnetic field B, per the formulation by American physicist Ernest O. Lawrence 1932 Berkeley California (Nobel Physics 1939 cyclotron invention). The classic non-relativistic formula: f_c = q · B / (2π · m), with q charge in coulombs (electron -e = 1.602×10⁻¹⁹ C; proton +e), B magnetic field in tesla, m rest mass kg (electron 9.109×10⁻³¹; proton 1.673×10⁻²⁷). Larmor radius (gyration): r_L = m · v_⊥ / (q · B). Angular velocity ω_c = q·B/m. For electron in Earth field B = 50 µT (Italy): f_c = 1.4 MHz. For electron B = 1 T neodymium magnet: f_c = 28 GHz. For proton B = 1.5 T MRI: f_c = 64 MHz (NMR Larmor). Relativistic correction: f_c_rel = f_c / γ Lorentz factor. Italian industrial-medical applications:
- MRI Esaote Genoa 1980 B = 1.5-3 T, f_proton = 64-128 MHz hospital imaging
- INFN Frascati 1957 DAΦNE φ-factory 1999 electrons-positrons 510 MeV, KLOE-2 K mesons
- CERN Geneva (Italian INFN Pisa Milan) LHC 27 km tunnel 13 TeV (Carlo Rubbia 1984 Nobel W Z)
- ENEA Frascati 1960 FTU 1990-2020 + DTT 2028 €600M (ICRF cyclotron heating 50-100 MHz protons)
- Consortium RFX Padua 1996 Reverse Field eXperiment (RF heating 7.2 MHz + NBI)
- Q-TOF FTICR mass spectrometers analytical chemistry. Italian astrophysics applications:
- Crab Pulsar PSR B0531+21 SN 1054 INAF Brera 1762 B = 4×10⁸ T relativistic synchrotron emission radio MHz-GHz
- Magnetar SGR 1806-20 B = 10¹⁰ T
- M87 EHT 2019 accretion disk synchrotron emission
- INAF Solar Arcetri 1872 heliosphere Parker Solar Probe pickup ions cyclotron heating
- Aurora ionosphere Italy 50 µT 1.4 MHz electrons. Carlo Rubbia 1984 Nobel CERN W Z bosons discovery
Italian cyclotron frequency tables
| Particle | q (C) | m (kg) | B (T) | f_c | Italian application |
|---|---|---|---|---|---|
| Electron | 1.602×10⁻¹⁹ | 9.109×10⁻³¹ | 50 µT (Earth) | 1.4 MHz | Aurora ionosphere Italy |
| Electron | 1.602×10⁻¹⁹ | 9.109×10⁻³¹ | 1 T (Nd) | 28 GHz | Gyrotron neodymium |
| Proton | 1.602×10⁻¹⁹ | 1.673×10⁻²⁷ | 1.5 T (MRI) | 64 MHz | MRI Esaote Genoa 1980 |
| Proton | 1.602×10⁻¹⁹ | 1.673×10⁻²⁷ | 3 T (MRI 3T) | 128 MHz | Advanced clinical MRI |
| Proton | 1.602×10⁻¹⁹ | 1.673×10⁻²⁷ | 5.4 T (DTT) | 83 MHz | ENEA Frascati DTT 2028 |
| Proton | 1.602×10⁻¹⁹ | 1.673×10⁻²⁷ | 8.33 T (LHC) | 127 MHz | CERN INFN 27 km tunnel |
| Deuterium ²H | 1.602×10⁻¹⁹ | 3.344×10⁻²⁷ | 5.4 T | 41 MHz | DTT fusion plasma |
| Tritium ³H | 1.602×10⁻¹⁹ | 5.008×10⁻²⁷ | 5.4 T | 27 MHz | D-T fusion plasma |
| Helium He²⁺ | 3.204×10⁻¹⁹ | 6.645×10⁻²⁷ | 1 T | 7.7 MHz | α-particle cancer therapy |
(Source: Lawrence 1932 Phys. Rev. cyclotron + NIST CODATA 2018 + Jackson "Classical Electrodynamics" Wiley 1999 + INFN Frascati 1957 + ENEA Frascati 1960).
Italian cyclotron frequency examples
- Clinical MRI 1.5 T Esaote Genoa 1980 hospital Policlinico Umberto I Rome 1903: water tissue protons, B = 1.5 T. f_c = 1.602×10⁻¹⁹ × 1.5 / (2π × 1.673×10⁻²⁷) = 63.87 MHz Larmor. Brain heart abdomen imaging 0.5-2 mm spatial resolution. Esaote Italy market leader 30%
- DTT Divertor Tokamak Test ENEA Frascati Rome 2028 €600M fusion plasma: B_toroidal = 5.4 T, plasma 10¹⁹ m⁻³ deuterium-tritium. f_c protons = 83 MHz, deuterons 41 MHz, tritons 27 MHz. ICRF Ion Cyclotron Resonance Frequency heating 50-100 MHz plasma to 100M K thermonuclear fusion
- DAΦNE INFN Frascati 1999 φ-factory electrons-positrons 510 MeV: B curving field 0.9 T 100 m diameter ring. Electrons γ ≈ 1,000 → f_c relativistic = 28 GHz / 1,000 = 28 MHz orbit
- LHC CERN Geneva 27 km tunnel 13 TeV protons: B = 8.33 T NbTi superconducting dipoles 1.9 K. γ_protons = 7,000 → f_c = 127 MHz / 7,000 = 18 kHz orbit. Italian INFN Pisa-Milan-Padua-Bologna contribution
- Italy ionosphere aurora 100 km altitude B = 50 µT: atmospheric electrons f_c = 1.4 MHz radio AKR Auroral Kilometric Radiation emission
- Crab Pulsar PSR B0531+21 SN 1054 INAF Brera 1762: rotating neutron star 31 ms period, B_surface ≈ 4×10⁸ T (10⁴× neodymium). f_c electrons > 10²⁰ Hz synchrotron gamma emission
- FTICR Fourier Transform Ion Cyclotron Resonance mass spectrometer Italian chemistry labs: B = 7-15 T static superconducting magnet. Mass resolution 10⁶-10⁷ Δm/m proteomics + metabolomics
Italian accelerators and plasmas
Italian particle accelerators:
- INFN Frascati National Laboratories 1957 (Rome, founder Edoardo Amaldi 1908-1989 'Via Panisperna boys' Fermi): DAΦNE φ-factory 1999 electrons-positrons 510 MeV, 97 m diameter ring, KLOE-2 K mesons + Siddharta exotic atoms; BTF Beam Test Facility 25-510 MeV electron beam line
- INFN LNGS Gran Sasso 1987 (Aquila, 1,400 m below Apennines, world's largest underground lab 180,000 m³): XENONnT 8.4 ton xenon WIMP dark matter; CRESST CaWO4 cryogenic mK; DarkSide-50 argon TPC; CUORE Te 0νββ; Borexino solar neutrinos CNO; OPERA superluminal neutrinos (2011 controversy retracted)
- INFN Pisa CMS LHC CERN; INFN Milan ATLAS LHC; INFN Padua LHCb; INFN Bologna Belle II Japan SuperKEKB
- CERN Geneva 1954 with major Italian contribution (Carlo Rubbia director 1989-1993 Nobel 1984 W Z bosons co-discovery UA1 1983; Lucio Rossi LHC magnet design; Fabiola Gianotti director 2016-present, two terms): LHC Large Hadron Collider 27 km tunnel B = 8.33 T NbTi 1.9 K liquid helium, 13 TeV proton-proton 2015-present, Higgs boson discovery 2012 ATLAS+CMS Nobel 2013 Englert-Higgs
- Italian fusion plasma: ENEA Frascati Rome 1960 FTU 1990-2020 (B = 8 T compact, plasma 10⁷ K); DTT 2028 (€600M, B = 5.4 T, 25 m³ plasma, liquid metal lithium divertor); Consortium RFX Padua 1996 Reverse Field eXperiment + SPIDER ITER NBI injector + MITICA (1 MV negative beam); ITER Cadarache France 2025-2035 35 countries €20B Italian ENEA contribution
- Carlo Rubbia 1984 Nobel Physics 'discovery field particles W Z weak interaction' CERN UA1 1983 (with Simon van der Meer stochastic cooling); Italian Physics Nobel Prizes: Marconi 1909 (radio), Fermi 1938 (fission), Segrè 1959 (antiproton Berkeley), Carlo Rubbia 1984 (W Z CERN), Riccardo Giacconi 2002 (X-ray astronomy). Italian R&D tax bonuses 2025: R&D Tax Credit L. 160/2019 up to 25%; Patent Box L. 178/2020; PNRR Mission 4 Component 2 €6.9B
Updated July 2026
Frequently Asked Questions
What is the cyclotron frequency?
Charged particle rotation frequency in uniform magnetic field. Formulated Ernest O. Lawrence 1932 Berkeley California (Nobel Physics 1939 cyclotron invention). Non-relativistic formula f_c = q·B/(2π·m) with q charge coulombs, B magnetic field tesla, m mass kg. Angular velocity ω_c = q·B/m. Larmor radius r_L = m·v_⊥/(q·B). Relativistic correction f_c_rel = f_c/γ Lorentz factor.
Cyclotron frequency formula?
f_c = q · B / (2π · m) where q particle charge coulombs (electron -e=1.602×10⁻¹⁹ C; proton +e), B magnetic field tesla, m rest mass kg (electron 9.109×10⁻³¹; proton 1.673×10⁻²⁷). Angular velocity ω_c = 2π·f_c = q·B/m. Larmor (gyration) radius r_L = m·v_⊥/(q·B). Relativistic f_c_rel = f_c/γ with γ = 1/√(1-v²/c²) Lorentz factor.
1.5 T clinical MRI example?
Clinical 1.5 T MRI Esaote Genoa 1980 hospital Policlinico Umberto I Rome 1903: water tissue protons, B = 1.5 T static field. f_c = 1.602×10⁻¹⁹ × 1.5 / (2π × 1.673×10⁻²⁷) = 63.87 MHz Larmor frequency. Brain heart abdomen imaging 0.5-2 mm spatial resolution. Esaote Italy market leader 30% MRI clinical hospital.
DTT 2028 fusion example?
DTT Divertor Tokamak Test ENEA Frascati Rome 2028 €600M thermonuclear fusion: B_toroidal = 5.4 T, plasma 10¹⁹ m⁻³ deuterium-tritium. f_c protons = 83 MHz, deuterons 41 MHz, tritons 27 MHz. ICRF Ion Cyclotron Resonance Frequency heating 50-100 MHz accelerates plasma to 100M K. ITER Cadarache 2025-2035 35 countries €20B Italian ENEA + RFX Padua contribution.
Italian particle accelerators?
Italian accelerators: INFN Frascati 1957 DAΦNE φ-factory 1999 electrons-positrons 510 MeV (B = 0.9 T, 97 m ring, KLOE-2); INFN Gran Sasso 1987 1,400 m below Apennines (XENONnT, CRESST, Borexino); CERN Geneva 1954 LHC 27 km B = 8.33 T 13 TeV (Carlo Rubbia director 1989-1993 Nobel 1984 W Z; Fabiola Gianotti 2016-present; Higgs 2012 ATLAS-CMS Nobel 2013).
Carlo Rubbia CERN Nobel?
Carlo Rubbia Gorizia 1934 Nobel Physics 1984 shared Simon van der Meer 'discovery field particles W Z weak interaction'. CERN UA1 collaboration 1983 SPS Super Proton Synchrotron 540 GeV W± Z⁰ bosons discovery (weak interaction mediators). CERN director general 1989-1993. Italian senator for life 2013-present. Lucia Votano LNGS director; Fabiola Gianotti CERN director 2016.
Italian cyclotron applications?
Italian cyclotron applications: 1) MRI Esaote Genoa 1980 1.5-3 T 64-128 MHz; 2) DAΦNE INFN Frascati 1999 φ-factory; 3) CERN LHC 27 km Italian INFN; 4) DTT ENEA Frascati 2028 fusion plasma; 5) RFX Padua 1996 reverse field; 6) FTICR mass spectrometers analytical chemistry; 7) Aurora ionosphere 1.4 MHz; 8) Crab Pulsar INAF Brera 4×10⁸ T synchrotron gamma; 9) Magnetar SGR 1806-20.
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