.. _fit-oscillation: Oscillation =========== A coherent muon-spin precession signal appears whenever the muon ensemble experiences a well-defined local field — an applied transverse field, or a spontaneous internal field set up by magnetic order. The components here are *undamped*: damping is intentionally separated, so a physical line shape is built by multiplying with a relaxation envelope, .. code-block:: text Oscillatory * Exponential + Constant for a Lorentzian-broadened line (dynamic disorder, dilute static moments) or .. code-block:: text Oscillatory * Gaussian + Constant for a Gaussian-broadened line (dense static field distribution). A bare ``Oscillatory + Constant`` fits only a perfectly coherent signal and will absorb the inevitable line shape into spurious phase and frequency residuals. When a signal contains several inequivalent muon sites, a sum of two or three ``Oscillatory`` components is generally preferable to one component with a broadened envelope; if the field distribution is genuinely continuous, look at the Fourier spectrum first (:doc:`../fourier_analysis`), and for an *incommensurate* distribution use ``Bessel``. .. _fit-oscillatory: Oscillatory ----------- .. math:: A(t) = A\,\cos(2\pi f t + \phi) Coherent precession parameterised by frequency. In zero field on an ordered magnet the spontaneous frequency :math:`f = \gamma_\mu B_{\mathrm{int}}/2\pi` acts as an order parameter and is the natural quantity to trend versus temperature (:doc:`../parameter_trending`); in transverse field the frequency calibrates the local field, :math:`f\,[\mathrm{MHz}] \simeq 0.01355\,B\,[\mathrm{G}]`. ============= ============ ===== ========================================== Name Symbol Unit Description ============= ============ ===== ========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``frequency`` :math:`f` MHz Precession frequency. ``phase`` :math:`\phi` rad Phase offset. ============= ============ ===== ========================================== The frequency is bounded non-negative; the phase is unrestricted. In a well-tuned spectrometer the phase of the first component should sit close to 0; large fitted phases usually indicate an instrumental phase offset that should be calibrated out, or a wrong model. Frequencies approaching the inverse binned time step alias: seed ``frequency`` from a Fourier peak (:doc:`../fourier_analysis`) rather than letting the minimiser search. A fit window covering fewer than two or three periods leaves the phase uncertain at the :math:`\sim\pi` level — extend the range or fix the phase. A standalone damped-cosine ``Oscillatory`` model (with ``Lambda`` and ``baseline``) is available in the ``MODELS`` registry. **References** - S. J. Blundell, Contemp. Phys. **40**, 175 (1999). .. _fit-oscillatory-field: OscillatoryField ---------------- .. math:: A(t) = A\,\cos(\gamma_\mu B\,t + \phi) The same precession parameterised by the local field :math:`B` (Gauss), with :math:`f = \gamma_\mu B/2\pi`. Use when the physically interesting quantity is the field itself — extracting the temperature dependence of a sublattice magnetisation, or comparing internal fields across runs in a parameter-trending workflow. For a transverse-field muonium experiment, model the central diamagnetic Mu⁺ line with this component and add ``MuoniumTF`` for the Mu⁰ satellites. Mathematically equivalent to ``Oscillatory``. ========= ============ ===== ============================================== Name Symbol Unit Description ========= ============ ===== ============================================== ``A`` :math:`A` % Component asymmetry amplitude. ``field`` :math:`B` G Local magnetic field at the muon site. ``phase`` :math:`\phi` rad Phase offset. ========= ============ ===== ============================================== .. _fit-bessel: Bessel ------ .. math:: A(t) = A\,J_0(2\pi f t + \phi) The polarisation of an **incommensurate** magnet, such as a spin-density-wave state. When the ordering wavevector is incommensurate with the lattice, the implanted muons uniformly sample the phase of the modulation and hence the Overhauser distribution of local fields, :math:`p(B) = \pi^{-1}(B_1^2 - B^2)^{-1/2}` for :math:`|B| < B_1`; the resulting polarisation is the zeroth-order Bessel function with :math:`f = \gamma_\mu B_1/2\pi` set by the field-distribution edge. At late times .. math:: J_0(x) \simeq \sqrt{\tfrac{2}{\pi x}}\,\cos(x - \tfrac{\pi}{4}), a damped cosine with a characteristic :math:`-45^\circ` phase — so a free-phase ``Oscillatory`` fit that insists on a phase near :math:`-45^\circ` is the classic sign that this component is needed. Compose with a relaxation envelope for additional damping; for commensurate order use ``Oscillatory`` or ``OscillatoryField``. ============= ============ ===== ========================================== Name Symbol Unit Description ============= ============ ===== ========================================== ``A`` :math:`A` % Component asymmetry amplitude. ``frequency`` :math:`f` MHz Field-distribution edge, γ\ :sub:`μ`\ B₁/2π. ``phase`` :math:`\phi` rad Phase offset. ============= ============ ===== ========================================== **References** - L. P. Le *et al.*, Phys. Rev. B **48**, 7284 (1993). .. _fit-vortex-lattice: VortexLattice / VortexLatticePowder ----------------------------------- .. math:: A(t) = A\,\mathrm{Re}\!\left[e^{i(2\pi\gamma_\mu B t + \phi)}\, R_{VL}(t;\lambda,B_{c2})\right] Transverse-field precession in the **mixed state of a type-II superconductor**. Below :math:`T_c` the muon samples the inhomogeneous field of the flux-line lattice, whose distribution :math:`p(B)` is strongly **non-Gaussian** — a sharp low-field cutoff at the saddle point, a most-probable field below the mean, and a long tail to high field near the vortex cores (a positively skewed line). The relaxation :math:`R(t)=\langle e^{i 2\pi\gamma_\mu(B-\bar B)t}\rangle` is the characteristic function of the *modified-London* field distribution of an ideal triangular lattice. Fitting the lineshape directly — rather than a single ``Gaussian`` proxy, whose returned rate depends on the fit window and binning — gives a window-independent penetration depth :math:`\lambda` and upper critical field :math:`B_{c2}`. ``VortexLatticePowder`` applies the :math:`3^{1/4}\lambda_{ab}` polycrystalline average and returns the ab-plane depth :math:`\lambda_{ab}`. The line's second moment is calibrated to the Brandt result (see :doc:`../sc_penetration_depth`), so the depth read from this lineshape matches the field-domain ``SC_Brandt_VortexLattice`` trend models. Multiply by a ``Gaussian`` for the nuclear dipolar background and add ``Oscillatory + Constant`` for the (weakly relaxing) sample-holder signal: .. code-block:: text VortexLatticePowder * Gaussian + Oscillatory + Constant ============== =================== ===== ===================================== Name Symbol Unit Description ============== =================== ===== ===================================== ``A`` :math:`A` % Component asymmetry amplitude. ``field`` :math:`B` G Applied transverse field (usually fixed). ``phase`` :math:`\phi` rad Phase offset. ``lambda_ab`` :math:`\lambda` nm Penetration depth (ab-plane for powder). ``Bc2`` :math:`B_{c2}` T Upper critical field (core-size cutoff). ============== =================== ===== ===================================== ``field`` starts fixed at the applied value. :math:`B_{c2}` is weakly constrained by a single low-field run (where :math:`b=B/B_{c2}\to 0`); fix it from the literature or fit the field dependence to pin it. ``lambda_ab`` is strongly correlated with the nuclear ``Gaussian`` rate, so constrain the latter from a normal-state (above :math:`T_c`) measurement at the same field. **References** - E. H. Brandt, Phys. Rev. B **68**, 054506 (2003). - J. E. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. **72**, 769 (2000). - F. L. Pratt *et al.*, Phys. Rev. B **79**, 052508 (2009).