Muonium

Muonium (Mu = μ⁺e⁻) forms when the implanted muon captures an electron, and its spin dynamics are governed by the hyperfine coupling \(A_{\mathrm{hf}}\) (4463 MHz for vacuum muonium; reduced in semiconductors and molecules). In a field \(B\) the four energy levels follow the Breit–Rabi diagram with reduced field

\[x = \frac{B}{B_0}, \qquad B_0 = \frac{A_{\mathrm{hf}}}{\gamma_e + \gamma_\mu} \;(\approx 1585\ \mathrm{G\ for\ vacuum\ Mu}),\]

and the observable transverse-field transitions \(\nu_{ij}\) carry amplitudes \((1 \pm \delta)/4\) with \(\delta = x/\sqrt{1+x^2}\). All components on this page compute their frequencies from the exact Breit–Rabi levels, use the positive-frequency (same-phase) convention, and exclude the central diamagnetic Mu⁺ line — model that separately with OscillatoryField. Compose with a relaxation envelope for damping.

MuoniumTF

\[A(t) = \frac{A}{4}\sum_{ij}(1\pm\delta)\cos(2\pi\nu_{ij} t + \phi), \qquad \nu_{ij} = E_i - E_j\]

The general transverse-field muonium component: all four hyperfine transitions (\(\nu_{12}\), \(\nu_{23}\), \(\nu_{14}\), \(\nu_{34}\)) with their Breit–Rabi amplitudes, parameterised by field \(B\) and hyperfine coupling \(A_{\mathrm{hf}}\). In the shallow-donor (small \(A_{\mathrm{hf}}\)) limit it reduces to two satellites straddling the diamagnetic line with separation \(A_{\mathrm{hf}}\), so the hyperfine constant can be read off directly.

Name	Symbol	Unit	Description
A	\(A\)	%	Muonium asymmetry amplitude.
field	\(B\)	G	Applied transverse field.
A_hf	\(A_{\mathrm{hf}}\)	MHz	Hyperfine coupling constant.
phase	\(\phi\)	rad	Phase offset.

For genuinely shallow-donor satellites it is often more robust to fit three independent Oscillatory lines with linked frequencies; this component targets genuine muonium where the relative transition weights matter.

References

• 2. 4. Patterson, Rev. Mod. Phys. 60, 69 (1988).

MuoniumLowTF

\[A(t) = \frac{A}{4}\left[(1+\delta)\cos(2\pi\nu_{12} t + \phi) + (1-\delta)\cos(2\pi\nu_{23} t + \phi)\right]\]

The low-field approximation: only the two intratriplet transitions \(\nu_{12}\) and \(\nu_{23}\), which are the lines observable at low transverse field (\(x \ll 1\)), where the other two transitions sit near \(A_{\mathrm{hf}}\) and are beyond the spectrometer bandwidth. Use when only the low-frequency pair is resolved; otherwise prefer MuoniumTF. Parameters as for MuoniumTF.

References

• 2. 4. Patterson, Rev. Mod. Phys. 60, 69 (1988).

MuoniumZF

\[A(t) = \frac{A}{6}\sum_k a_k\cos(2\pi f_k t + \phi), \qquad f_1 = A_{\mathrm{hf}} - D,\quad f_2 = A_{\mathrm{hf}} + \tfrac{D}{2},\quad f_3 = \tfrac{3D}{2}\]

Zero-field muonium with an axially anisotropic hyperfine interaction: three lines set by the isotropic coupling \(A_{\mathrm{hf}}\) and the axial component \(D\), with weights \((1, 2, 2)/6\) and an optional Lorentzian cutoff f_cut suppressing lines beyond the spectrometer bandwidth. There is no applied field, so no diamagnetic line. Relevant for anisotropic muonium centres (e.g. bond-centred Mu in semiconductors) measured in zero field.

Name	Symbol	Unit	Description
A	\(A\)	%	Muonium asymmetry amplitude.
A_hf	\(A_{\mathrm{hf}}\)	MHz	Isotropic hyperfine coupling.
D_mu	\(D\)	MHz	Axial hyperfine anisotropy.
f_cut	\(f_{\mathrm{cut}}\)	MHz	Lorentzian cutoff (0 = off).
phase	\(\phi\)	rad	Phase offset.

References

• 2. 4. Patterson, Rev. Mod. Phys. 60, 69 (1988).

MuoniumHighTF

\[A(t) = \frac{A}{2}\left[\cos(2\pi\nu_{12} t + \phi) + \cos(2\pi\nu_{34} t + \phi)\right], \qquad \nu_{12} + \nu_{34} = A_{\mathrm{hf}}\]

The high transverse-field muonium pair. Above \(B_0\) only the two muon-spin-flip transitions survive (the \(1-\delta\) amplitudes vanish as \(\delta \to 1\)), and their frequencies sum to the hyperfine constant — so fitting the pair measures \(A_{\mathrm{hf}}\) directly even when neither line is individually assigned. The equal \(1/2\) weights are the high-field limit; at lower fields, where the amplitudes differ, use MuoniumTF. Parameters as for MuoniumTF.

References

• 2. 4. Patterson, Rev. Mod. Phys. 60, 69 (1988).

MuoniumHighTFAniso

\[A(t) = \frac{A}{2}\left\langle \cos\!\left[2\pi\nu_{12}(\theta)\,t+\phi\right] + \cos\!\left[2\pi\nu_{34}(\theta)\,t+\phi\right] \right\rangle_{\cos\theta}, \qquad \nu_{12}(\theta) + \nu_{34}(\theta) \simeq A_{\mathrm{hf}} + \tfrac{D}{2}\left(3\cos^2\theta - 1\right)\]

The high-TF pair with an axially anisotropic hyperfine interaction, powder averaged: writing the hyperfine tensor as an isotropic part \(A_{\mathrm{hf}}\) plus an axial (traceless) part \(D\), the two muon-spin-flip frequencies are obtained for each crystallite orientation by exact diagonalisation of the 4-level Hamiltonian \(H = \gamma_e B S_z^e - \gamma_\mu B S_z^\mu + S^e\!\cdot\!A(\theta)\!\cdot\!S^\mu\), batched over a 32-node Gauss–Legendre \(\cos\theta\) grid. Both lines co-shift so that each orientation’s pair sum tracks the secular effective coupling \(A_{\mathrm{eff}}(\theta) = A_{\mathrm{hf}} + \tfrac{D}{2}(3\cos^2\theta - 1)\), producing the characteristic asymmetric (Pake-like) powder broadening. Use for bond-centred muonium in semiconductors or muoniated radicals in powders; \(D = 0\) reduces exactly to MuoniumHighTF, and for single crystals fit the orientation-dependent lines directly. (WiMDA’s AnisMuoniumPairRot instead splits its two signed line frequencies by a symmetric \(\pm d/2\), which is only approximate — fitted \(D\) values are not directly comparable.)

Name	Symbol	Unit	Description
A	\(A\)	%	Muonium asymmetry amplitude.
field	\(B\)	G	Applied transverse field.
A_hf	\(A_{\mathrm{hf}}\)	MHz	Isotropic hyperfine coupling.
D_mu	\(D\)	MHz	Axial hyperfine anisotropy.
phase	\(\phi\)	rad	Phase offset.

References

• 2. 4. Patterson, Rev. Mod. Phys. 60, 69 (1988).

• 5. Roduner and H. Fischer, Chem. Phys. 54, 261 (1981).

MuoniumLFRelax

\[A(t) = A\,e^{-\lambda t}, \qquad \lambda = \frac{(1-\delta)\,\delta_{ex}^2\,\tau_c} {1 + (2\pi\nu_{12}\tau_c)^2}, \qquad \delta = \frac{x}{\sqrt{1+x^2}}\]

Longitudinal-field spin-lattice (T₁) relaxation of muonium by a fluctuating coupling — nuclear hyperfine fields modulated by muonium hopping, or electron spin exchange with carriers — sampled at the intratriplet \(\nu_{12}\) transition in the BPP/Redfield form used throughout the muonium quantum-diffusion literature. The \((1-\delta)\) prefactor quenches the relaxation as the muon repolarises in high field; together with the growing \(\nu_{12}\) this produces the LF quenching curves from which hop rates are extracted. Measuring \(\lambda\) versus \(B\) and locating the T₁ minimum (\(2\pi\nu_{12}\tau_c \approx 1\)) determines both \(\delta_{ex}\) and \(\tau_c\).

Name	Symbol	Unit	Description
A	\(A\)	%	Relaxing amplitude.
delta_ex	\(\delta_{ex}\)	MHz	Fluctuating-coupling amplitude.
tau_c	\(\tau_c\)	μs	Correlation time.
B_L	\(B_L\)	G	Applied longitudinal field.
A_hf	\(A_{\mathrm{hf}}\)	MHz	Hyperfine coupling (normally fixed).

\(\nu_{12}\) is computed from the exact Breit–Rabi levels — intentionally not WiMDA’s approximate expression (see docs/porting/wimda-fit-function-parity/), so fitted \(\delta_{ex}\)/\(\tau_c\) are not directly comparable with WiMDA’s. A_hf defaults to vacuum muonium and should normally stay fixed. This is a relaxation envelope: multiply an oscillating component, or use it standalone for the repolarised muonium fraction.

References

• 18. 6. Kiefl et al., Phys. Rev. Lett. 62, 792 (1989).

• 18. Kadono et al., Phys. Rev. Lett. 64, 665 (1990).