.. _diffusion-ballistic-lf: 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 :math:`A(t)`. The standard workflow is two-stage: fit each longitudinal-field (LF) decoupled run to an empirical relaxation form to extract :math:`\lambda(B_{LF})`, then fit the resulting :math:`\lambda` versus :math:`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 :doc:`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 :math:`S(t)` of the fluctuating local field is built from the appropriate transport propagator, its one-sided cosine transform gives the spectral density, .. math:: 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, .. math:: \lambda(B_{LF}) \;=\; \frac{A^2}{4}\,J(\omega_e), \qquad \omega_e = \gamma_e B_{LF}. The amplitude :math:`A` is the coupling strength of the muon to the fluctuating field, in :math:`\mathrm{MHz}` (numerically equal to :math:`\mu s^{-1}` here). The Larmor frequency is taken from the electron gyromagnetic ratio :math:`\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 :math:`n` accessible dimensions, following Pratt [2], is .. math:: 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 :math:`I_0` the modified Bessel function of the first kind. :math:`D_{nD}` is the in-plane (or in-chain) diffusion rate; :math:`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 :math:`\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, .. math:: \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 :doc:`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`` :math:`A` MHz Coupling amplitude (numerically equal to μs⁻¹). ``D_nD`` :math:`D_{nD}` μs⁻¹ In-plane / in-chain diffusion rate. ``D_perp`` :math:`D_\perp` μs⁻¹ Perpendicular diffusion rate. ``B_LF`` :math:`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 :math:`n` dimensions follows the Bessel-function form derived in *Muon Spectroscopy: An Introduction* [3], .. math:: S_{nD}^{\mathrm{ball}}(t) \;=\; \bigl[J_0(2 D_{\mathrm{hop}}\,t)\bigr]^{2n}, \quad n \in \{1,2,3\}, with :math:`J_0` the zeroth-order Bessel function of the first kind and :math:`D_{\mathrm{hop}}` the ballistic hopping rate. The spectral-density construction is identical to the diffusive case, and the field-dependent rate is again .. math:: \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, .. math:: J(\omega) \;\approx\; \frac{0.318}{D_{\mathrm{hop}}} \,\ln\!\left(\frac{16 D_{\mathrm{hop}}}{\omega}\right), so that :math:`\lambda` plotted against :math:`\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`` :math:`A` MHz Coupling amplitude. ``D_hop`` :math:`D_{\mathrm{hop}}` μs⁻¹ Ballistic hopping rate. ``B_LF`` :math:`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 :math:`\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 :math:`\lambda` versus :math:`B_{LF}` series extracted from a previous round of time-domain fits. The end-to-end workflow is documented in :doc:`parameter_trending`; the call signature for direct evaluation is .. code-block:: python 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: .. code-block:: python 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 :math:`A` collapses the whole coupling to one number and the Larmor frequency is taken from :math:`\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 :math:`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 :math:`\chi^2` to discriminate them reliably; fix the dimensionality from the crystallography. - **The 1D ballistic log-form is a low-frequency approximation.** The :math:`\lambda \propto \ln B_{LF}` linearity holds only over the field window where :math:`\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.