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,

Oscillatory * Exponential + Constant

for a Lorentzian-broadened line (dynamic disorder, dilute static moments) or

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 (Fourier analysis), and for an incommensurate distribution use Bessel.

Oscillatory

\[A(t) = A\,\cos(2\pi f t + \phi)\]

Coherent precession parameterised by frequency. In zero field on an ordered magnet the spontaneous frequency \(f = \gamma_\mu B_{\mathrm{int}}/2\pi\) acts as an order parameter and is the natural quantity to trend versus temperature (Parameter trending); in transverse field the frequency calibrates the local field, \(f\,[\mathrm{MHz}] \simeq 0.01355\,B\,[\mathrm{G}]\).

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
frequency	\(f\)	MHz	Precession frequency.
phase	\(\phi\)	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 (Fourier analysis) rather than letting the minimiser search. A fit window covering fewer than two or three periods leaves the phase uncertain at the \(\sim\pi\) 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

• 19. 10. Blundell, Contemp. Phys. 40, 175 (1999).

OscillatoryField

\[A(t) = A\,\cos(\gamma_\mu B\,t + \phi)\]

The same precession parameterised by the local field \(B\) (Gauss), with \(f = \gamma_\mu B/2\pi\). 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	\(A\)	%	Component asymmetry amplitude.
field	\(B\)	G	Local magnetic field at the muon site.
phase	\(\phi\)	rad	Phase offset.

Bessel

\[A(t) = A\,J_0(2\pi f t + \phi)\]

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, \(p(B) = \pi^{-1}(B_1^2 - B^2)^{-1/2}\) for \(|B| < B_1\); the resulting polarisation is the zeroth-order Bessel function with \(f = \gamma_\mu B_1/2\pi\) set by the field-distribution edge. At late times

\[J_0(x) \simeq \sqrt{\tfrac{2}{\pi x}}\,\cos(x - \tfrac{\pi}{4}),\]

a damped cosine with a characteristic \(-45^\circ\) phase — so a free-phase Oscillatory fit that insists on a phase near \(-45^\circ\) 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	\(A\)	%	Component asymmetry amplitude.
frequency	\(f\)	MHz	Field-distribution edge, γμB₁/2π.
phase	\(\phi\)	rad	Phase offset.

References

• 12. 16. Le et al., Phys. Rev. B 48, 7284 (1993).

VortexLattice / VortexLatticePowder

\[A(t) = A\,\mathrm{Re}\!\left[e^{i(2\pi\gamma_\mu B t + \phi)}\, R_{VL}(t;\lambda,B_{c2})\right]\]

Transverse-field precession in the mixed state of a type-II superconductor. Below \(T_c\) the muon samples the inhomogeneous field of the flux-line lattice, whose distribution \(p(B)\) 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 \(R(t)=\langle e^{i 2\pi\gamma_\mu(B-\bar B)t}\rangle\) 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 \(\lambda\) and upper critical field \(B_{c2}\).

VortexLatticePowder applies the \(3^{1/4}\lambda_{ab}\) polycrystalline average and returns the ab-plane depth \(\lambda_{ab}\). The line’s second moment is calibrated to the Brandt result (see Superconducting penetration depth models), 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:

VortexLatticePowder * Gaussian + Oscillatory + Constant

Name	Symbol	Unit	Description
A	\(A\)	%	Component asymmetry amplitude.
field	\(B\)	G	Applied transverse field (usually fixed).
phase	\(\phi\)	rad	Phase offset.
lambda_ab	\(\lambda\)	nm	Penetration depth (ab-plane for powder).
Bc2	\(B_{c2}\)	T	Upper critical field (core-size cutoff).

field starts fixed at the applied value. \(B_{c2}\) is weakly constrained by a single low-field run (where \(b=B/B_{c2}\to 0\)); 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 \(T_c\)) measurement at the same field.

References

• 5. 8. Brandt, Phys. Rev. B 68, 054506 (2003).

• 10. 5. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. 72, 769 (2000).

• 6. 12. Pratt et al., Phys. Rev. B 79, 052508 (2009).