Pauli Particles in 3D

Pauli's Hamiltonian for an electron of charge $-1$ is

\[H_{\text{Pauli}} = \tfrac{1}{2} \, (p - A(x))^{2} - \dfrac{\hbar}{2} \, \sigma \cdot B(x) + \phi (x) ,\]

where $\sigma = (\sigma_{x}, \sigma_{y}, \sigma_{z})$ is the vector of Pauli matrices describing the intrinsic magnetic moment observed in the Stern-Gerlach experiment. In most regimes of classical physics that intrinsic moment is negligible. Surprisingly, however, introducing a formal magnetic moment term $\mu \vert B \vert$ into the Hamiltonian of a classical particle opens the way to structure-preserving integrators for guiding centre dynamics. This is the classical Pauli particle,

\[H (x,p) = \tfrac{1}{2} \, (p - A(x))^{2} + \mu \vert B(x) \vert + \phi (x) ,\]

with corresponding Lagrangian

\[L (x, \dot{x}) = \tfrac{1}{2} \vert \dot{x} \vert^{2} + A(x) \cdot \dot{x} - \mu \vert B(x) \vert - \phi (x) .\]

The construction follows Jianyuan Xiao and Hong Qin, Slow manifolds of classical Pauli particle enable structure-preserving geometric algorithms for guiding center dynamics, arXiv:2006.03818 (2020).

Relation to the Other Models

The Pauli particle sits between the charged particle and the guiding centre:

Model	Lagrangian
charged particle	$L = \tfrac{1}{2} \vert \dot{x} \vert^{2} + A(x) \cdot \dot{x} - \phi (x)$
classical Pauli particle	$L = \tfrac{1}{2} \vert \dot{x} \vert^{2} + A(x) \cdot \dot{x} - \mu \vert B(x) \vert - \phi (x)$
guiding centre	$L = ( A(x) + u \, b(x) ) \cdot \dot{x} - \tfrac{1}{2} u^{2} - \mu \vert B(x) \vert - \phi (x)$, with $u = \dot{x} \cdot b(x)$

It differs from the charged particle only by the $\mu \vert B \vert$ term, so it retains the full six-dimensional phasespace and a regular Lagrangian — unlike the guiding centre model, whose Lagrangian is degenerate. That is the point: the Pauli particle can be integrated with ordinary symplectic or variational partitioned Runge-Kutta methods, with no projection onto a constraint manifold.

Slow Manifolds

Solutions of the classical Pauli particle with (almost) no gyration, $\dot{x} \times b(x) \approx 0$, can be viewed as slow manifolds of the Pauli dynamics. If

\[\mu' = \dfrac{\vert \dot{x} \times b(x) \vert^{2}}{2 \vert B(x) \vert} \approx 0 ,\]

the gyro radius is close to zero and the particle position nearly coincides with its guiding centre. From this perspective, guiding centre dynamics is the slow manifold of the classical Pauli particle, and a structure-preserving integrator for the latter inherits the good long-time behaviour on the former.

See also Joshua W. Burby, Guiding center dynamics as motion on a formal slow manifold in loop space, Journal of Mathematical Physics 61, 012703 (2020), doi:10.1063/1.5119801.

Implementation

The models are written in the same curvilinear form as the other families, with the one-form

\[\vartheta_{i} = g_{ij} (x) \, v^{j} + A_{i} (x)\]

and the Hamiltonian

\[H = \tfrac{1}{2} \, g_{ij} (x) \, v^{i} v^{j} + \mu \vert B(x) \vert + \varphi (x) ,\]

where $g_{ij}$ is the (diagonal) metric of the coordinate system and $\mu$ is carried as a problem parameter. Mass and charge are normalised to one; see Normalization.

initial_conditions(x₀, v₀) splits a velocity vector into the part parallel to the magnetic field and the magnetic moment of the perpendicular part,

\[u_{0} = v_{0} \cdot b (x_{0}) , \qquad v_{\parallel} = u_{0} \, b (x_{0}) , \qquad v_{\perp} = v_{0} - v_{\parallel} , \qquad \mu = \dfrac{\vert v_{\perp} \vert^{2}}{2 \vert B (x_{0}) \vert} ,\]

and returns (q = x₀, v = v∥, params = (μ = μ,)), which every problem constructor of this module takes directly — podeproblem(initial_conditions(x₀, v₀)). See Initialization for the route from an energy and a pitch angle instead.

The problem is available in partitioned (podeproblem), Hamiltonian (hodeproblem) and implicit (iodeproblem) form.

Usage

using GeometricIntegrators
using ChargedParticleDynamics.PauliParticle3d.TokamakSmallCylindrical

problem  = hodeproblem(initial_conditions_pauli())
solution = integrate(problem, PartitionedGauss(1))

For the small axisymmetric tokamak in cylindrical coordinates, with

\[A (R, Z, \varphi) = \frac{B_{0} R_{0}}{2} \, \frac{Z}{R} \, \mathrm{d} R - \frac{B_{0} R_{0}}{2} \, \ln \bigg( \frac{R}{R_{0}} \bigg) \, \mathrm{d} Z - \frac{B_{0} r^{2}}{2 q_{0}} \, \mathrm{d} \varphi , \qquad B_{0} = 1 , \quad R_{0} = 1 , \quad q_{0} = 2 ,\]

the reference initial condition of the charged particle,

\[x_{0} = [ 1.05, \, 0, \, 0 ] , \qquad v_{0} = [ 2.1 \times 10^{-3}, \, 0, \, 4.3 \times 10^{-4} ] ,\]

corresponds to

\[v_{\parallel} \approx [ 0, \, 1.074 \times 10^{-5}, \, 4.297 \times 10^{-4} ] , \qquad \mu \approx 2.315 \times 10^{-6} ,\]

which is the magnetic moment default_parameters() returns for that equilibrium, and to the guiding centre initial condition $q_{0} = [x_{0}, u_{0}]$ with $u_{0} = v_{0} \cdot b (x_{0}) \approx 4.299 \times 10^{-4}$. The three models can therefore be compared directly on the same physical orbit — and scripts/study_model_agreement.jl measures how directly.

Which components these are

The two velocities above are quoted in physical components, $\sqrt{g_{ii}} \, v^{i}$, which is what makes their third entries read $4.3 \times 10^{-4}$ rather than the contravariant $4.095 \times 10^{-4}$ the module actually stores — the chart has $g_{33} = R^{2}$ and $R = 1.05$. And $u_{0}$ is the contraction $v_{0} \cdot b$ with the covariant b, not $\vert v_{\parallel} \vert$: those agree to three digits here only because the parallel velocity is almost entirely toroidal. The three triads b, bₚ and b⃗ are covariant, physical and contravariant and are not interchangeable; see Initialization.

Both signs above were negative in this page before ElectromagneticFields 0.7.0, and one of the two changes was a correction rather than an update. $v_{\parallel}$ and $\mu$ are invariant under the field reversal — $v_{\parallel} = (v \cdot b) \, \vec{b}$ flips twice — so their signs were simply wrong. $u_{0}$ genuinely was negative in this chart, because $b$ pointed the wrong way; it is positive now, and agrees with the cartesian chart of the same tokamak, which it did not before. See "The reversed field of the left-handed charts" in Findings.

Modules

ChargedParticleDynamics.PauliParticle3d — Module

The Pauli particle in three dimensions: the charged particle plus the $\mu \vert B \vert$ term of the guiding centre, on the full six-dimensional phasespace.

It sits between the charged particle and the guiding centre — it keeps a regular Lagrangian, so it needs no projection onto a constraint manifold. See the Pauli Particles in 3D page.