Field-dependent transport models (DiffusionLF / BallisticLF)

When muon-spin relaxation is dominated by motion of the muon (or its host spin environment) rather than by a static field distribution, the diagnostic observable is the field dependence of the relaxation rate rather than the detailed shape of any single \(A(t)\). The standard workflow is two-stage: fit each longitudinal-field (LF) decoupled run to an empirical relaxation form to extract \(\lambda(B_{LF})\), then fit the resulting \(\lambda\) versus \(B_{LF}\) curve with a transport model whose parameters carry direct physical meaning — a diffusion rate, a hopping rate, the dimensionality of the motion. Asymmetry implements two families of such models, both consumed by the parameter-trending framework documented in Parameter trending:

DiffusionLF_1D / 2D / 3D for incoherent random-walk transport, in one, two, or three spatial dimensions.
BallisticLF_1D / 2D / 3D for coherent ballistic transport at the same set of dimensionalities.

Both families share a common construction: an autocorrelation function \(S(t)\) of the fluctuating local field is built from the appropriate transport propagator, its one-sided cosine transform gives the spectral density,

\[J(\omega) \;=\; 2\int_0^{\infty} S(t)\,\cos(\omega t)\,dt,\]

and the muon relaxation rate is read out at the longitudinal Larmor frequency,

\[\lambda(B_{LF}) \;=\; \frac{A^2}{4}\,J(\omega_e), \qquad \omega_e = \gamma_e B_{LF}.\]

The amplitude \(A\) is the coupling strength of the muon to the fluctuating field, in \(\mathrm{MHz}\) (numerically equal to \(\mu s^{-1}\) here). The Larmor frequency is taken from the electron gyromagnetic ratio \(\gamma_e\) because the field that the muon sees in these systems is typically set by an unpaired electron spin on a host atom.

Choosing between the two families is a physics question, not a fitting question: are you describing a random walk (diffusion) or coherent motion along well-defined channels (ballistic)? In a strongly disordered or strongly scattered system the random-walk picture is correct and the DiffusionLF_* models apply; in a clean, low-dimensional conductor with long mean free path the coherent-transport picture is correct and the BallisticLF_* models apply. Dimensionality is the other axis of choice and is usually fixed by the crystal structure of the host: one-dimensional chains, two-dimensional layers, or three-dimensional bulk.

Diffusive transport (DiffusionLF_*)

The autocorrelation for incoherent diffusion in \(n\) accessible dimensions, following Pratt [2], is

\[S_{nD}(t) \;=\; \bigl[e^{-2 D_{nD} t}\,I_0(2 D_{nD} t)\bigr]^{n} \,\bigl[e^{-2 D_\perp t}\,I_0(2 D_\perp t)\bigr]^{3-n}, \quad n \in \{1,2,3\},\]

with \(I_0\) the modified Bessel function of the first kind. \(D_{nD}\) is the in-plane (or in-chain) diffusion rate; \(D_\perp\) allows for a slower perpendicular diffusion channel and may be fixed to zero in genuinely low-dimensional cases. Substituting into the spectral-density expression above gives the field-dependent rate \(\lambda_{nD}^{\mathrm{diff}}(B_{LF})\).

The full parameter-trending expression that the documentation in [1] employs for a quasi-2D conductor is a sum of four physically distinct contributions,

\[\lambda(B_{LF}) \;=\; \lambda_{nD}^{\mathrm{diff}}(B_{LF}) + \lambda_{0D}(B_{LF}) + \lambda_{BG} + \lambda_{LCR}(B_{LF}),\]

a dynamic transport term, a local (0D) dynamic term (the Redfield component in Parameter trending), a field-independent background (Lambda_bg), and an optional level-crossing-resonance Gaussian (GaussianLCR). Asymmetry exposes each of these as a separate basis function in the parameter-model builder so the user can include only the terms motivated by their data.

Parameters

Name	Symbol	Unit	Description
`A`	\(A\)	MHz	Coupling amplitude (numerically equal to μs⁻¹).
`D_nD`	\(D_{nD}\)	μs⁻¹	In-plane / in-chain diffusion rate.
`D_perp`	\(D_\perp\)	μs⁻¹	Perpendicular diffusion rate.
`B_LF`	\(B_{LF}\)	G	Longitudinal field (independent variable).

The dimensionality is encoded in the choice of component (1D, 2D, or 3D) rather than as a fit parameter. D_perp may be fixed at zero for cleanly low-dimensional systems; freeing it lets the fit accommodate weak inter-chain or inter-layer leakage.

Ballistic transport (BallisticLF_*)

For coherent propagation, the autocorrelation in \(n\) dimensions follows the Bessel-function form derived in Muon Spectroscopy: An Introduction [3],

\[S_{nD}^{\mathrm{ball}}(t) \;=\; \bigl[J_0(2 D_{\mathrm{hop}}\,t)\bigr]^{2n}, \quad n \in \{1,2,3\},\]

with \(J_0\) the zeroth-order Bessel function of the first kind and \(D_{\mathrm{hop}}\) the ballistic hopping rate. The spectral-density construction is identical to the diffusive case, and the field-dependent rate is again

\[\lambda_{nD}^{\mathrm{ball}}(B_{LF}) \;=\; \frac{A^2}{4}\,J(\omega_e), \quad \omega_e = \gamma_e B_{LF}.\]

The one-dimensional ballistic case has a useful low-frequency approximation,

\[J(\omega) \;\approx\; \frac{0.318}{D_{\mathrm{hop}}} \,\ln\!\left(\frac{16 D_{\mathrm{hop}}}{\omega}\right),\]

so that \(\lambda\) plotted against \(\log B_{LF}\) is expected to be close to linear over the corresponding field window. Curvature relative to that prediction is usually the cleanest indicator that motion is not strictly one-dimensional or not strictly ballistic.

Parameters

Name	Symbol	Unit	Description
`A`	\(A\)	MHz	Coupling amplitude.
`D_hop`	\(D_{\mathrm{hop}}\)	μs⁻¹	Ballistic hopping rate.
`B_LF`	\(B_{LF}\)	G	Longitudinal field (independent variable).

Choosing dimensionality

The dimensionality choice should be made from the crystallography and any prior transport measurements on the same compound, not from comparing fit qualities across the three options.

*LF_1D for transport along chains (e.g. organic conductor stacks) or other strongly one-dimensional pathways.
*LF_2D for layered systems with in-plane mobility (e.g. cuprates, intercalates, transition-metal dichalcogenides).
*LF_3D when the transport is bulk-like and isotropic on the muon time scale.

A fit that selects BallisticLF_1D over DiffusionLF_2D purely on \(\chi^2\) is rarely conclusive on its own: the field dependence over any realistic experimental window is similar enough between dimensionalities that the choice usually rests on the physics. Where the data genuinely support discrimination, the model-selection workflow in the parameter-trending fits (AIC / BIC) will report it; treat that as confirmation rather than as a primary tool.

Using these components

These components live in the parameter-model registry (PARAMETER_MODEL_COMPONENTS) and are applied to a \(\lambda\) versus \(B_{LF}\) series extracted from a previous round of time-domain fits. The end-to-end workflow is documented in Parameter trending; the call signature for direct evaluation is

import numpy as np
from asymmetry.core.fitting.parameter_models import PARAMETER_MODEL_COMPONENTS

b_lf = np.linspace(10.0, 3000.0, 60)  # Gauss

diff = PARAMETER_MODEL_COMPONENTS["DiffusionLF_2D"]
lam_diff = diff.function(b_lf, A=0.8, D_nD=2.0, D_perp=0.0)

ball = PARAMETER_MODEL_COMPONENTS["BallisticLF_1D"]
lam_ball = ball.function(b_lf, A=0.8, D_hop=3.0)

In a composite parameter model, combine the transport term with the field-independent background and (if needed) the level-crossing-resonance term:

from asymmetry.core.fitting.parameter_models import ParameterCompositeModel

model = ParameterCompositeModel(
    ["DiffusionLF_2D", "Lambda_bg"], operators=["+"]
)

The GUI parameter-model builder filters the available components by the selected x-axis so that field-only components are not offered for temperature trends, and vice versa.

Assumptions and limitations

Both families are idealised transport propagators, and the fitted parameters should be read with the following caveats in mind:

Each model describes one limiting regime. DiffusionLF_* assumes purely incoherent random-walk motion and BallisticLF_* assumes purely coherent propagation; a real system that sits between the two — or crosses over between them as temperature or field changes — is not captured exactly by either, and a good fit to one form does not by itself establish that regime.
A single dominant, static-on-average fluctuating coupling is assumed. The amplitude \(A\) collapses the whole coupling to one number and the Larmor frequency is taken from \(\gamma_e\) because the field is assumed to be set by a single unpaired electron spin on a host atom; systems with several comparable couplings, or where a nuclear rather than electronic moment dominates, need a different construction.
Disorder, trapping, and finite-size effects break the ideal propagators. Both autocorrelation forms assume an unbounded, homogeneous lattice with a fixed dimensionality; carrier trapping, strong quenched disorder, or motion on a finite cluster all modify \(S(t)\) in ways the closed forms do not represent.
The dimensionality is an input, not an output. As noted above, the field dependence over a realistic experimental window is often too similar between dimensionalities (and between the diffusive and ballistic families) for \(\chi^2\) to discriminate them reliably; fix the dimensionality from the crystallography.
The 1D ballistic log-form is a low-frequency approximation. The \(\lambda \propto \ln B_{LF}\) linearity holds only over the field window where \(\omega \ll 16 D_{\mathrm{hop}}\); outside it the full spectral density must be used.

References

[1] F. L. Pratt, F. Lang, W. Steinhardt, S. Haravifard, and S. J.: Blundell, Phys. Rev. B 106, L060401 (2022) — the four-term decomposition used in the quasi-2D dynamic-relaxation analysis the diffusive components were built to match.
[2] F. L. Pratt, J. Phys.: Conf. Ser. 2462, 012038 (2023) —: autocorrelation forms for one-, two-, and three-dimensional diffusion.
[3] S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt,: Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022) — derivation of the ballistic Bessel-function autocorrelation.