7 Spin Tracking
This section describes how OPALX evolves the per-particle spin polarization vector along the trajectory. The model is the Thomas-BMT equation (Bargmann et al. 1959; Jackson 1999), integrated as a closed-form Rodrigues rotation so that the magnitude \(|\vec P|\) is preserved exactly per step.
7.1 Polarization vector
OPALX stores, for each tracked particle, a three-component polarization vector \[ \vec P = \big(\langle\hat\sigma_x\rangle,\; \langle\hat\sigma_y\rangle,\; \langle\hat\sigma_z\rangle\big), \tag{7.1}\] where each component is the expectation value of the corresponding Pauli operator. The expectation is evaluated in the particle’s instantaneous (Lorentz) rest frame, while the three Cartesian axes along which it is projected are the lab-frame axes. This hybrid convention is the one the BMT equation is naturally written in: the spin state lives in the rest frame, but the rotation rate \(\vec\Omega\) is expressed using the lab-frame fields \(\vec E, \vec B\) at the particle’s position, so keeping the projection axes lab-aligned avoids a redundant boost rotation inside the integrator.
The magnitude \(|\vec P|\) is bounded by the spin-\(\tfrac{1}{2}\) Bloch sphere, \[ 0 \le |\vec P| \le 1 , \tag{7.2}\] with \(|\vec P|=1\) corresponding to a fully polarized state and \(|\vec P|=0\) to an unpolarized state.
7.2 The Thomas-BMT equation
In the lab frame, with \(t\) the lab-frame time, \(\vec P\) evolves as \[ \frac{d\vec P}{dt} = \vec\Omega \times \vec P, \tag{7.3}\] where the angular velocity \(\vec\Omega\) for a particle of charge \(q\), rest mass \(m\), and gyromagnetic anomaly \(G = (g-2)/2\) in electromagnetic fields \(\vec E, \vec B\) is \[ \vec\Omega = -\frac{q}{m}\Bigg[ \Big(G + \frac{1}{\gamma}\Big)\,\vec B - \frac{G\,\gamma}{\gamma + 1}\, \big(\vec\beta \cdot \vec B\big)\,\vec\beta - \Big(G + \frac{1}{\gamma + 1}\Big)\, \frac{\vec\beta \times \vec E}{c} \Bigg]. \tag{7.4}\] Here \(\vec\beta = \vec v / c\) is the lab-frame velocity in units of \(c\) and \(\gamma = (1 - \beta^2)^{-1/2}\) is the corresponding Lorentz factor. The derivation is given in (Jackson 1999, sec. 11.11); we use the standard sign convention for \(\vec\Omega\) such that Equation 7.3 describes the time evolution of \(\vec P\) rather than that of a rotating frame.
Term-by-term, Equation 7.4 separates into three physically distinct contributions:
Cyclotron-Larmor term, \(-\tfrac{q}{m}\,(G + \tfrac{1}{\gamma})\vec B\). The \(1/\gamma\) piece is the relativistic Larmor precession at the cyclotron frequency \(\omega_c = qB/(\gamma m)\), common to any charged particle in a magnetic field. The \(G\) piece is the additional precession driven by the anomalous magnetic moment — it is the part the BMT equation contributes beyond the Dirac result \(g = 2\).
Longitudinal-\(B\) correction, \(+\tfrac{q}{m}\,\tfrac{G\gamma}{\gamma+1}\,(\vec\beta\cdot\vec B)\vec\beta\). This subtracts the component of \(\vec B\) along the motion that is ineffective at rotating \(\vec P\). It vanishes when \(\vec\beta \perp \vec B\) (the canonical storage-ring geometry).
Motional-electric term, \(+\tfrac{q}{m}\,(G + \tfrac{1}{\gamma+1})\,(\vec\beta\times\vec E)/c\). This is the BMT response to electric fields. Even at \(G = 0\) (Dirac) it does not vanish — the residual \(1/(\gamma+1)\) is the Thomas-precession piece, which is a kinematic effect of the successive Lorentz boosts and survives in the absence of anomalous moments.
Setting \(G = 0\) in Equation 7.4 reduces the rotation to pure Larmor precession of a Dirac particle: \(\vec\Omega = -(q/m)\,(\vec B/\gamma)\), i.e. at \(\omega_c\) when \(\vec\beta \perp \vec B\). Any deviation of \(g\) from \(2\) — which is generic for composite particles and even for QED loop-corrected leptons — adds the \(G\vec B\) contribution that physically defines the BMT regime.
OPALX currently tabulates \(G\) for the species that have a well-defined classical anomaly:
| Species | \(G = (g-2)/2\) | Source |
|---|---|---|
| Electron, positron | \(1.15965218128 \times 10^{-3}\) | QED (Jackson 1999) |
| Muon (\(\mu^\pm\)) | \(1.16592061 \times 10^{-3}\) | (Particle Data Group 2024) |
| Proton, antiproton | \(1.792847386\) | (Jackson 1999) |
Species not listed default to \(G = 0\). In the current OPALX release only the muon channel is actively spin-tracked downstream; extending the integrator itself to another species is just a matter of populating \(G\) from the species table.
7.3 OPALX unit conventions
OPALX stores rest mass as the rest energy in eV (numerically \(m\,c^2\)), and charge in proton-charge units (dimensionless, e.g. \(-1\) for \(\mu^-\), \(+1\) for \(\mu^+\)). With these conventions, the SI ratio \(q/m\) that appears in Equation 7.4 becomes \[ \frac{q}{m}\bigg|_{\text{SI}} \;=\; \frac{\text{charge}\,\cdot\,c^{2}}{\text{mass}_{\text{eV}}} , \tag{7.5}\] because the elementary-charge factor cancels between numerator (units of \(e\)) and denominator (units of eV). This is the same idiom used by the Boris pusher (Birdsall and Langdon 1985) in OPALX; the spin and momentum updates therefore consume \(q, m\) in identical internal forms with no separate unit conversion needed.
7.4 Numerical integration: Rodrigues rotation
A naive explicit ODE solver applied to Equation 7.3 (e.g. RK4) does not preserve \(|\vec P|\) and accumulates a slow magnitude drift over time. OPALX therefore integrates Equation 7.3 as a closed-form rotation.
For a single step of duration \(\Delta t\) during which \(\vec\Omega\) is held constant, the exact solution is a rotation of \(\vec P\) about \(\hat n \equiv \vec\Omega / |\vec\Omega|\) by angle \[ \phi = |\vec\Omega|\,\Delta t . \tag{7.6}\] The rotation is evaluated through Rodrigues’ formula, \[ \vec P' = \cos\phi\;\vec P + \sin\phi\,(\hat n \times \vec P) + (1 - \cos\phi)\,(\hat n \cdot \vec P)\,\hat n . \tag{7.7}\] By construction this preserves \(|\vec P|\) to machine precision per step, independent of \(\phi\), so the magnitude drift typical of explicit non-symplectic schemes is eliminated.
The remaining accuracy concern is the constant-\(\vec\Omega\) assumption across the step. For small per-step rotations the error incurred by holding \(\vec\Omega\) frozen at its value at the start of the step is third order in \(\phi\). To bound this error robustly when the beamline contains strong fields or when \(\Delta t\) is large compared to the spin-precession period, OPALX sub-steps whenever the per-step rotation exceeds a threshold \(\phi_{\max} = 0.1\) rad: the timestep is split into \[ N_{\text{sub}} = \left\lceil \frac{\phi}{\phi_{\max}} \right\rceil \tag{7.8}\] equal sub-rotations of angle \(\phi / N_{\text{sub}}\) each, with \(\vec\Omega\) re-evaluated identically for each (since \(\vec E, \vec B, \gamma\) do not change inside a single tracker step). The threshold \(\phi_{\max} = 0.1\) rad bounds the rotation-truncation error at the \(10^{-4}\) relative level per step, which is well below the float storage precision of \(\vec P\).
7.5 Placement in the tracker step
The polarization update is applied after both external and space-charge contributions to \(\vec E\) and \(\vec B\) have been accumulated at each particle, but before the momentum kick that closes the time step. This ordering guarantees that the \(\vec E, \vec B\) used to construct \(\vec\Omega\) in Equation 7.4 are precisely the fields the particle experiences during the step — the same fields the Boris kick uses — so the momentum and spin halves of the step see a consistent electromagnetic environment, and no extra mid-step field-interpolation is required.