.. _fit-muonium: Muonium ======= Muonium (Mu = μ⁺e⁻) forms when the implanted muon captures an electron, and its spin dynamics are governed by the hyperfine coupling :math:`A_{\mathrm{hf}}` (4463 MHz for vacuum muonium; reduced in semiconductors and molecules). In a field :math:`B` the four energy levels follow the Breit–Rabi diagram with reduced field .. math:: 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 :math:`\nu_{ij}` carry amplitudes :math:`(1 \pm \delta)/4` with :math:`\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 :ref:`fit-oscillatory-field`. Compose with a relaxation envelope for damping. .. _fit-muonium-tf: MuoniumTF --------- .. math:: 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 (:math:`\nu_{12}`, :math:`\nu_{23}`, :math:`\nu_{14}`, :math:`\nu_{34}`) with their Breit–Rabi amplitudes, parameterised by field :math:`B` and hyperfine coupling :math:`A_{\mathrm{hf}}`. In the shallow-donor (small :math:`A_{\mathrm{hf}}`) limit it reduces to two satellites straddling the diamagnetic line with separation :math:`A_{\mathrm{hf}}`, so the hyperfine constant can be read off directly. ========== ======================= ===== ==================================== Name Symbol Unit Description ========== ======================= ===== ==================================== ``A`` :math:`A` % Muonium asymmetry amplitude. ``field`` :math:`B` G Applied transverse field. ``A_hf`` :math:`A_{\mathrm{hf}}` MHz Hyperfine coupling constant. ``phase`` :math:`\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** - B. D. Patterson, Rev. Mod. Phys. **60**, 69 (1988). .. _fit-muonium-low-tf: MuoniumLowTF ------------ .. math:: 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 :math:`\nu_{12}` and :math:`\nu_{23}`, which are the lines observable at low transverse field (:math:`x \ll 1`), where the other two transitions sit near :math:`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** - B. D. Patterson, Rev. Mod. Phys. **60**, 69 (1988). .. _fit-muonium-zf: MuoniumZF --------- .. math:: 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 :math:`A_{\mathrm{hf}}` and the axial component :math:`D`, with weights :math:`(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`` :math:`A` % Muonium asymmetry amplitude. ``A_hf`` :math:`A_{\mathrm{hf}}` MHz Isotropic hyperfine coupling. ``D_mu`` :math:`D` MHz Axial hyperfine anisotropy. ``f_cut`` :math:`f_{\mathrm{cut}}` MHz Lorentzian cutoff (0 = off). ``phase`` :math:`\phi` rad Phase offset. ========== ========================= ===== ================================== **References** - B. D. Patterson, Rev. Mod. Phys. **60**, 69 (1988). .. _fit-muonium-high-tf: MuoniumHighTF ------------- .. math:: 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 :math:`B_0` only the two muon-spin-flip transitions survive (the :math:`1-\delta` amplitudes vanish as :math:`\delta \to 1`), and their frequencies sum to the hyperfine constant — so fitting the pair measures :math:`A_{\mathrm{hf}}` directly even when neither line is individually assigned. The equal :math:`1/2` weights are the high-field limit; at lower fields, where the amplitudes differ, use ``MuoniumTF``. Parameters as for ``MuoniumTF``. **References** - B. D. Patterson, Rev. Mod. Phys. **60**, 69 (1988). .. _fit-muonium-high-tf-aniso: MuoniumHighTFAniso ------------------ .. math:: 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 :math:`A_{\mathrm{hf}}` plus an axial (traceless) part :math:`D`, the two muon-spin-flip frequencies are obtained for each crystallite orientation by **exact diagonalisation of the 4-level Hamiltonian** :math:`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 :math:`\cos\theta` grid. Both lines co-shift so that each orientation's pair sum tracks the secular effective coupling :math:`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; :math:`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 :math:`\pm d/2`, which is only approximate — fitted :math:`D` values are not directly comparable.) ========== ======================= ===== ==================================== Name Symbol Unit Description ========== ======================= ===== ==================================== ``A`` :math:`A` % Muonium asymmetry amplitude. ``field`` :math:`B` G Applied transverse field. ``A_hf`` :math:`A_{\mathrm{hf}}` MHz Isotropic hyperfine coupling. ``D_mu`` :math:`D` MHz Axial hyperfine anisotropy. ``phase`` :math:`\phi` rad Phase offset. ========== ======================= ===== ==================================== **References** - B. D. Patterson, Rev. Mod. Phys. **60**, 69 (1988). - E. Roduner and H. Fischer, Chem. Phys. **54**, 261 (1981). .. _fit-muonium-lf-relax: MuoniumLFRelax -------------- .. math:: 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 :math:`\nu_{12}` transition in the BPP/Redfield form used throughout the muonium quantum-diffusion literature. The :math:`(1-\delta)` prefactor quenches the relaxation as the muon repolarises in high field; together with the growing :math:`\nu_{12}` this produces the LF quenching curves from which hop rates are extracted. Measuring :math:`\lambda` versus :math:`B` and locating the T₁ minimum (:math:`2\pi\nu_{12}\tau_c \approx 1`) determines both :math:`\delta_{ex}` and :math:`\tau_c`. ============ ======================= ===== ================================== Name Symbol Unit Description ============ ======================= ===== ================================== ``A`` :math:`A` % Relaxing amplitude. ``delta_ex`` :math:`\delta_{ex}` MHz Fluctuating-coupling amplitude. ``tau_c`` :math:`\tau_c` μs Correlation time. ``B_L`` :math:`B_L` G Applied longitudinal field. ``A_hf`` :math:`A_{\mathrm{hf}}` MHz Hyperfine coupling (normally fixed). ============ ======================= ===== ================================== :math:`\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 :math:`\delta_{ex}`/:math:`\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** - R. F. Kiefl *et al.*, Phys. Rev. Lett. **62**, 792 (1989). - R. Kadono *et al.*, Phys. Rev. Lett. **64**, 665 (1990).