Physics-informed X-ray tomography of magnetization and material parameters

Technical Summary: Physics-informed X-ray Tomography of Magnetization and Material Parameters

Problem Statement
Three-dimensional magnetic textures, such as skyrmion strings, hopfions, and vortex rings, are stabilized by microscopic interactions within host materials. While X-ray magnetic vector tomography (using X-ray magnetic circular dichroism, or XMCD) can image these nanoscale structures, the technique faces significant limitations. Weak magnetic contrast often results in noisy projections, and practical constraints on synchrotron beamtime, radiation dose, and accessible measurement angles frequently lead to incomplete data (e.g., missing wedges or cones). Consequently, standard iterative reconstructions often leave nanoscale features poorly constrained or introduce artifacts, particularly when the experimental resolution approaches the scale of the magnetic features themselves. Furthermore, traditional approaches that apply micromagnetic models only after reconstruction cannot recover features lost during the imaging process and rely on time-consuming manual tuning.

Methodology
The authors propose a "physics-informed" reconstruction framework that integrates the micromagnetic energy functional directly into the tomographic optimization process. Instead of treating the reconstruction as a purely mathematical inverse problem, the method minimizes a joint objective function $L(m)$:
$$L(m) = L^2_{\text{data}}(m) + \lambda_E E_{\text{total}}(m)$$
where $L^2_{\text{data}}$ measures the misfit between simulated and measured XMCD projections, and $E_{\text{total}}$ represents the total micromagnetic energy of the sample. This energy term includes exchange stiffness ($A$), demagnetizing interactions (scaled by saturation magnetization $M_s$), and the Dzyaloshinskii–Moriya interaction (DMI, constant $D$).

The reconstruction utilizes a gradient-based optimizer (Riemannian Adam) with automatic differentiation to update the magnetization vector field $\mathbf{m}$ (where $|\mathbf{m}|=1$). The forward model simulates the line integrals of the magnetization component along the beam direction for dual-axis tilt series. The micromagnetic energy acts as a physical prior, constraining the solution to physically plausible states without blurring domain walls, as the full energy functional (including competing interactions) is preserved rather than just a smoothing term.

Key Contributions and Results
The method was applied to a 1.1 µm-diameter, 300 nm-thick disk of the room-temperature chiral magnet $\text{Co}_8\text{Zn}_9\text{Mn}_3$, imaged using dichroic soft X-ray ptychography.

1. Recovery from Sparse and Single-Axis Data:
   The study demonstrates that the physics-informed approach can recover the full 3D helical texture from significantly reduced datasets.

   • Sparse Sampling: Accurate reconstructions were achieved using only 9 projections (every 8th projection from a standard dual-axis set) compared to the full 67.
   • Single-Axis Tomography: The method successfully reconstructed the 3D magnetization using data from a single tilt axis (either the $xz$ or $yz$ series) alone. While a single axis cannot measure the magnetization component parallel to the rotation axis, the micromagnetic energy couples the unmeasured component to the measured ones, allowing the optimizer to infer the missing information consistent with the material's physics.
   • Validation: Using a synthetic ground truth generated from the experimental data, the authors showed that without the physics prior, the error in in-plane components was high ($\epsilon_m = 0.74$). With the balanced physics prior, the error dropped to $\epsilon_m = 0.11$, accurately recovering all three vector components.

2. Parameter Sensitivity and Readout:
   The authors investigated whether the reconstruction could determine material parameters ($A$, $D$, $M_s$) by analyzing the data misfit at high energy weights ($\lambda_E$).

   • Sensitivity: The data misfit was found to be relatively insensitive to $M_s$ but highly sensitive to the ratio of exchange stiffness to DMI ($A/D$).
   • Readout Value: By minimizing the data misfit across a grid of parameters, the method identified an optimal DMI constant of $D = 0.66 \times 10^{-3} \text{ J m}^{-2}$ (assuming literature $A_0$). This value is approximately 10% higher than the literature value ($D_0 = 0.60 \times 10^{-3} \text{ J m}^{-2}$) derived from the bulk helical period relation ($L_D = 4\pi A/D$).
   • Explanation: The discrepancy is attributed to the difference between the bulk relation (valid for unbounded crystals) and the confined geometry of the nanostructure. The physics-informed model, which accounts for the full energy of the confined disk, yields a parameter set that better reproduces the measured texture than the bulk-derived value.

Significance and Claims
The paper claims that this approach offers a dual benefit for magnetic imaging:

1. Efficiency: It reduces the time and beamtime required for vector tomography by enabling robust 3D reconstructions from sparse data or single-axis acquisitions, facilitating time-resolved and field-dependent studies.
2. Material Characterization: It extends the utility of tomography beyond imaging to "texture-driven magnetometry," allowing for the determination of fundamental magnetic interaction parameters (specifically the $A/D$ ratio) directly from the nanostructure's texture.

The authors emphasize that the method is modular; the physics prior can be paired with any differentiable forward model (e.g., electron holography or Lorentz tomography) and extended to include other material interactions, provided the forward model is differentiable. The work suggests a strategy where a single, well-sampled measurement of a reference state can validate the energy model and fix parameters, allowing subsequent conditions to be imaged with sparse acquisitions.