Dynamic loop of a thin magnetic disk in an AC field
A thin disk (diameter D, thickness t) is treated as one spin: its shape anisotropy confines the moment to the disk plane, a weak uniaxial anisotropy Ku selects the in-plane easy axis x, and the alternating field H(t) = H0 sin(2πf t) is applied along x. The page solves the Fokker–Planck equation for the in-plane angle φ and shows the dynamic loop mx(H), the dissipated energy per cycle, and the quantities relevant for magnetic particle imaging (point-spread-function width) and hyperthermia (power density).
Everything on this page is computed live in your browser (Web Worker, no server, no lookup tables) from the one-spin model of a thin disk below; the Python reference solver gives identical numbers (area/(4B0) = 0.0214 for the default case).
Note: the same thin-disk model is now also available as a mode in /fp-loop/ (mode selector “thin disk”), next to the uniaxial and cubic particle models. This page stays as it is.
Companion applet: spherical nanoparticle with uniaxial anisotropy (Brown’s equation in θ) →
How the model works: energy, the unknown W(φ,t), the reduced equation, symbols, numerics, validityfor a physicist who has not read the paper
What is computed
A magnetic disk of diameter D and thickness t ≪ D, small enough to be uniformly magnetised, is treated as a single macrospin μ = MsV, V = πD²t/4. The disk is immobilised and does not interact with neighbours. Thermal agitation makes the moment direction fluctuate; the applet solves the Fokker–Planck equation for the probability density of the in-plane angle, live in your browser, and integrates the loop of the ensemble average mx(t) = ⟨cos φ⟩ over one period.
The energy
The shape anisotropy of a thin cylinder is an easy plane of strength K⊥ = μ0Ms²(Nz − Nx)/2, with Nx = (τ/π)(ln(4/τ) − 1/2), Nz = 1 − 2Nx, τ = t/D. It is so strong (μ0HK,⊥ = 2K⊥/Ms is of order 1 T) that the moment stays in the plane, m = (cosφ, sinφ, ε) with a small tilt ε. In the plane the energy density is
f(φ) = −Ku cos²φ − μ0Ms H(t) cosφ, H(t) = H0 sin(2πf t) along xFor Ku > 0 there are two wells, φ = 0 and φ = π, separated by the barrier KuV (reduced: σu = KuV/kBT) at zero field; the field tilts the landscape. For Ku = 0 the disk is a free planar rotor whose equilibrium magnetisation is I1(ξ)/I0(ξ), ξ = μ0MsVH/kBT (dashed in the loop plot, next to the 3-D Langevin function L(ξ) of an isotropic moment).
The unknown: W(φ, t)
W(φ, t) is the probability density of the in-plane angle: W dφ is the probability that the moment points within [φ, φ+dφ] of the easy axis at time t; it is periodic in φ and normalised, ∫02π W dφ = 1. W is not the magnetisation; the magnetisation along the field is its first harmonic mx(t) = ∫ cosφ W dφ. The heat map below shows W itself.
The reduced equation
Mobility in three lines. LLG in the angles, with γ' = γ/(1+α²):
ε' = −(γ'/Ms) [ 2αK⊥ε + f'(φ) ], φ' = −(γ'/Ms) [ α f'(φ) − 2K⊥ε ]The tilt relaxes fast, ε' ≈ 0 ⇒ 2K⊥ε = −f'/α; inserting gives φ' = −(γ'/Ms)(α + 1/α) f' = −(γ/(αMs)) f'(φ): the in-plane mobility is γ/(αMs), a factor (1+α²)/α² (101 for α = 0.1) larger than for an isotropic macrospin — the precession about the strong easy-plane field is converted into in-plane motion. With thermal noise this becomes the Fokker–Planck (Smoluchowski) equation
∂W/∂t = ∂/∂φ [ (γ/(αMs)) ( W ∂f/∂φ + (kBT/V) ∂W/∂φ ) ]Symbols
| Symbol | Meaning | Unit · how it enters |
|---|---|---|
| D, t, τ = t/D | disk diameter, thickness, aspect ratio | nm · volume and demagnetising factors |
| V = πD²t/4 | disk volume | m³ · thermal energy scale kBT/V |
| Ms | saturation magnetisation | A/m · Zeeman energy, K⊥, mobility, energy per cycle |
| Ku | in-plane uniaxial anisotropy (easy axis x) | J/m³ · −Ku cos²φ |
| Nx, Nz | thin-cylinder demagnetising factors | – · Nx = (τ/π)(ln(4/τ) − 1/2), Nz = 1 − 2Nx |
| K⊥, μ0HK,⊥ | easy-plane (shape) anisotropy and its field 2K⊥/Ms | J/m³, T · keeps the moment in the plane, validity limit |
| α, γ | Gilbert damping, gyromagnetic ratio 1.760859×1011 | –, rad s−1T−1 · mobility γ/(αMs) |
| T, kBT | temperature, thermal energy | K, J · diffusion term |
| H(t), H0, B0 = μ0H0, f | AC field along x, amplitude, frequency | A/m, mT, Hz · drive |
| φ | in-plane angle of the moment from x | rad · the single variable |
| W(φ, t) | probability density of φ | 1/rad · the solved quantity |
| mx = ∫ cosφ W dφ | reduced magnetisation along the field | – · loop mx(μ0H) |
| A = −∮ mx d(μ0H) | loop area | T · area/(4B0) = fraction of the ideal square loop |
| E = MsA, P = fE | energy per cycle and power density | J/m³, W/m³ · hyperthermia |
| σu = KuV/kBT | reduced in-plane barrier | – · blocking of the in-plane switching |
| kBT/m, 4.16 kBT/m | thermal field of the moment m = MsV; Langevin PSF FWHM estimate | mT · MPI resolution scale |
| Hc, Mr | |H| at mx = 0; mx at H = 0 | mT; units of Ms |
| α²B⊥/4 | validity limit of the overdamped reduction, B⊥ = μ0HK,⊥ | mT · compare with B0 |
Numerics — what “live” means
φ ∈ [0, 2π) is divided into N periodic finite-volume cells (default 256). Fluxes use the Scharfetter–Gummel scheme, so the discrete equilibrium is exactly the Boltzmann distribution. Time stepping is implicit Euler (L-stable, robust for the stiff high-field part) with S steps per period (default 4000); each step is a periodic (cyclic) tridiagonal solve, done with the Thomas algorithm plus a Sherman–Morrison correction. The periodic steady state comes from the propagator over half a period and the symmetry (φ, H) → (φ + π, −H) (a shift by N/2 cells): W* = R Mhalf W*, one dense N×N solve with one row replaced by the normalisation. From W* one period is propagated to record mx(t) and W(φ, t). Cost: about 0.6–1 s per loop in a Web Worker (shown in the status line).
Validity
Eliminating the tilt assumes that it relaxes faster than the in-plane motion. This holds strictly for in-plane fields B ≪ α²B⊥/4 (B⊥ = 2K⊥/Ms); for a 1-nm, 200-nm permalloy-like disk with α = 0.1 that is about 2.4 mT, i.e. the default 20 mT is outside the strict range and the card is flagged amber. Stochastic-LLG checks are running; at 300 kHz / 20 mT / Ku = 500 J/m³ the reduction agrees with micromagnetics within 5 %.
Steady-state loop mx(μ0H)
dynamic loopplanar rotor I₁/I₀ (equil., Ku=0)3-D Langevin L(ξ)Distribution W(φ, t)
The solved quantity: probability density of the in-plane angle over one field period; the loop above is its first harmonic ⟨cosφ⟩. Colour = log10 W; vertical axis φ (0 = along +x in the middle, ±180° = along −x at the edges); thin line = h(t)/h0 (+1 top, −1 bottom).