Cyclotron Resonance

Definition

Cyclotron resonance is what happens when you drive a gyrating charge at the exact frequency it already turns. Put a charged particle in a uniform magnetic field and the magnetic part of the Lorentz force bends its motion into a circle. The rate of that circular motion is the cyclotron frequency:

omega_c = qB / m        (angular, rad/s)
f_c     = qB / (2*pi*m)  (ordinary, Hz)

Now add a small oscillating electric field rotating in the plane of the orbit. If its frequency matches omega_c, it pushes the particle forward on every single loop — like timing your pushes on a child's swing. Energy accumulates, the orbit spirals outward, and the particle resonantly absorbs power. Mistune the drive and the push falls out of step, doing as much braking as accelerating, so nothing builds up.

How it works

Start with the Lorentz force on a particle of charge q moving with velocity v in field B:

F = q (v × B)

For motion perpendicular to B, this force is always perpendicular to v — the hallmark of circular motion. Setting the magnetic force equal to the centripetal requirement:

q v B = m v² / r
=>  v / r = q B / m
=>  omega_c = q B / m

The speed v cancels. This is the crucial, slightly magical fact: the gyration frequency is the same whether the particle crawls or races. A fast particle simply traces a bigger circle (radius r = mv/qB, the gyroradius) and arrives back at the start in the identical time. This isochronism is why the resonance is sharp, and why a single fixed-frequency drive can keep accelerating particles whose orbits grow from millimetres to metres.

The absorbed power is a textbook resonance. Treating gyration as a damped driven oscillator with collision rate 1/tau, the absorption is a Lorentzian peaked at omega = omega_c with quality factor Q ≈ omega_c · tau. You only get a clean, narrow peak when omega_c · tau >> 1 — the particle must complete many loops between collisions.

Worked example: the 28 GHz electron

Take a free electron in a B = 1 T field. Plug in q = 1.602×10⁻¹⁹ C and m = 9.109×10⁻³¹ kg:

omega_c = qB/m = (1.602e-19 × 1) / 9.109e-31
        = 1.759e11 rad/s

f_c = omega_c / (2*pi) = 2.80e10 Hz = 28.0 GHz

So an electron in a 1-tesla field circles 28 billion times a second — a microwave frequency. Scale linearly with B: at the 2.3 T used in many electron-cyclotron-resonance ion sources, electrons resonate near 64 GHz; at the ~5–6 T on-axis field of a fusion device, near 140–170 GHz, which is exactly the band of the gyrotrons that heat them.

A proton is 1836× heavier, so its cyclotron frequency is 1836× lower:

f_c(proton) = 28.0 GHz / 1836 ≈ 15.2 MHz per tesla

That is an ordinary shortwave-radio frequency — which is why ion cyclotron resonance heating uses RF antennas, not microwave dishes.

Now the orbit growth. Suppose the rotating drive field has amplitude E. On resonance the particle gains energy continuously; its perpendicular speed grows roughly linearly, v⊥ ≈ (qE/m)·t, and since r = mv⊥/qB the gyroradius widens as r ≈ (E/B)·t. A modest 1 kV/m drive in a 1 T field pumps the radius outward at about 1 km/s of radial growth — the spiral you see in the animation, just slowed down so the eye can follow it.

Measuring effective mass in solids

The single most useful laboratory application is reading off the effective mass m* of charge carriers in a crystal. Inside a solid, an electron does not respond to forces with its bare mass; the periodic lattice potential dresses it, and it accelerates as if it had a different mass set by the curvature of the energy band, d²E/dk². That dressed mass governs conductivity, heat capacity and how fast transistors switch — so measuring it matters.

The recipe, pioneered by Dresselhaus, Kip and Kittel on silicon and germanium in 1955:

1. Cool a high-purity crystal to a few kelvin and apply a strong, known magnetic field B.
2. Shine microwaves on it and sweep either the frequency or the field.
3. Find where absorption peaks. At the peak, omega = qB/m*, so m* = qB / (2*pi*f).

Because the band curvature differs along different crystal directions, m* is generally anisotropic: rotating the sample relative to B shifts the resonance, mapping out the full effective-mass tensor. The measurement only works in the clean limit omega_c·tau >> 1, which is precisely why it demands liquid-helium temperatures, defect-free samples, and the highest fields available. Modern versions in graphene and other 2D materials use the same idea to extract masses that can be a hundredth of the free-electron mass.

Variants and regimes

Regime / variant	Particle & frequency	What changes	Where it appears
Classical (non-relativistic)	omega_c = qB/m, fixed	Perfectly isochronous	Low-energy cyclotrons, lab CR
Relativistic	omega_c = qB/(gamma·m)	Frequency drops as energy rises	High-energy accelerators
Synchrocyclotron	Drive frequency swept down	Tracks gamma over the spiral	Proton therapy, 200+ MeV
Isochronous cyclotron	B shaped to rise with radius	Keeps qB/(gamma·m) constant	High-current beams, isotopes
Electron cyclotron (ECRH)	~28 GHz/T, here 100–170 GHz	Microwave gyrotron drive	Tokamaks, stellarators, ECR ion sources
Ion cyclotron (ICRH)	~15 MHz/T, tens of MHz	RF antenna drive	Tokamak ion heating
Solid-state CR	omega_c = qB/m*	Mass replaced by effective mass	Semiconductor metrology

Common pitfalls and misconceptions

• "Faster particles resonate at a higher frequency." No — that is the whole point. omega_c is independent of speed; faster particles just spiral wider. The frequency only changes once relativistic gamma kicks in.
• "Any oscillating field at omega_c will heat it." Only the circularly polarized component rotating in the same sense as the gyration does net work. The counter-rotating half mostly cancels. Pick the wrong handedness and you couple to holes instead of electrons.
• "It is the magnetic field that adds the energy." The magnetic force is always perpendicular to v, so it can do no work — it only bends. All the energy comes from the resonant electric field; B merely sets the tempo.
• "Stronger field, stronger absorption." Stronger B raises omega_c and shifts the resonance, but the absorbed power is governed by the drive amplitude and by omega_c·tau. A dirty, collision-dominated sample (omega_c·tau < 1) shows a broad, weak smear, not a sharp peak.
• "Cyclotron frequency equals Larmor frequency exactly." For orbital motion, yes, qB/m. For spin precession the g-factor enters; in plasma physics 'gyrofrequency' is the safe synonym.
• "You can heat a fusion plasma to any temperature this way." As ions get hot and relativistic, or as collisions thermalize the gyration phase, the resonance broadens and detunes, capping the efficiency at a given frequency.

Applications

• Fusion plasma heating. ECRH (electron cyclotron resonance heating) and ICRH (ion cyclotron resonance heating) deliver tens of megawatts into tokamaks and stellarators. Because B varies across the device and omega_c = qB/m, choosing the wave frequency selects exactly which radius gets heated — power steering by frequency.
• Effective-mass metrology. The reference technique for m* in semiconductors and 2D materials, including anisotropic and sub-percent-of-m_e masses.
• Particle accelerators. The original cyclotron (Lawrence, 1932) and its descendants accelerate protons and ions for physics, medicine (proton therapy) and isotope production.
• ECR ion sources. Microwaves at the electron cyclotron frequency strip atoms of many electrons, producing highly charged ion beams for accelerators and industry.
• Mass spectrometry. Fourier-transform ion cyclotron resonance (FT-ICR) measures mass-to-charge with extreme precision by detecting the cyclotron frequency of trapped ions.
• Space and astrophysics. Electron cyclotron maser emission lights up planetary auroral radio bursts; ion cyclotron waves shape the solar wind and magnetospheres.

Performance and derivation analysis

How sharp is the resonance, and how much can you absorb? Model the perpendicular velocity of a single particle as a driven, damped oscillator hit by a co-rotating field E·e^(−iωt):

m (dv/dt) = q E e^(-i*omega*t) - m v / tau + q (v × B)

Solving in steady state, the absorbed power per particle peaks at omega = omega_c with the Lorentzian form:

P(omega) ∝ (1/tau) / [ (omega - omega_c)² + (1/tau)² ]

The full width at half maximum is 2/tau, so the fractional linewidth is

Δomega / omega_c ≈ 2 / (omega_c · tau) = 2 / Q

This single dimensionless number omega_c·tau decides everything. In a fusion-grade plasma or an ultrapure semiconductor at a few kelvin it can reach 10²–10⁴, giving a knife-edge resonance you can use as a precision ruler. In a warm, collisional gas it dips below 1, smearing the peak into uselessness. The same physics that makes the resonance a measurement tool (narrow line) is what makes it a heating tool (large, selective power deposition): a narrow line means the energy goes exactly where omega matches qB/m, and nowhere else.

One more practical scaling. The energy gain per orbit is roughly q·E·(2*pi*r), proportional to radius, so the spiral accelerates: each loop adds more than the last and the radius grows linearly in time on resonance. That linear-in-time outward spiral is exactly what the interactive animation traces when the drive locks onto omega_c — and exactly what falls apart the moment you detune it.