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MuSR /
RadicalSimulation< Isotropic muonium diffusion | Index | Avoided Level Crossing (ALC) > We now proceed to simulate the Paschen-Back precessions of a radical in the liquid state, by matlab. The eigenvalues are obtained by the same routine given for muonium, the only difference being a reduced hyperfine frequency. The full precession pattern is produced by the following commented code:
However it is also interesting to generate the level diagram vs. field, as shown in the previous pages, or the amplitudes of the different polarization components. In particular we might be interested in reproducing the so-called longitudinal field repolarization curve starting from the simple case of isotropic muonium, in the absence of additional nuclear spins (e.g. for an all-carbon material such as fullerene or graphite). We want to determine the time dependent muon polarization in a longitudinal experiment, where the external field, the initial muon spin and the axis of the detectors lie along {$\hat z $}, which is given by {$P(t)=\frac 1 2 Tr\rho(t)I_z=Tr I_z(t)I_z$}. The Hamiltonian does not commute with {$I_z$}. Its eigenvalues are {$\hbar\omega_m$}, corresponding to eigenvectors {$| m \rangle$} with {$n=1,\dots,4$}. Therefore the muon time dependent polarization may be written as: {$ P(t)= \sum_{m,n} e^{-i\omega_m t}\langle m| I_z |n\rangle e^{i\omega_n t}\langle n|I_z |m\rangle $} The resulting time dependent function may be separated into two contributions, a time independent one that collects all diagonal terms ({$n=m$}), and a time dependent part that collects the precessing, non diagonal terms ({$n\ne m$}). The latter will oscillate at frequencies {$\omega_{nm}(\theta)=\omega_n(\theta)-\omega_m(\theta)$}, which may be observed for {$t>0$}, but, for large hyperfine interactions together with fast relaxation rates might escape detection. It is instead rather easy to detect the former contribution as the {$t=0$} extrapolation of the non precessing term, whose experimental asymmetry will therefore reflect the reduced polarization {$(1)\qquad\qquad P_R(H)=\sum_m (\langle m| I_z | m\rangle)^2 $} where the dependence on {$H$} is implicit in the eigenvectors {$|m(H)\rangle$} of the radical Hamiltonian. This leads to the following simple matlab simulation:
If we now want to simulate the repolarization curve of anisotropic muonium, still in the absence of additional nuclear spins, let us assume for simplicity that the radical anisotropic pseudodipolar hyperfine interaction {$ \hbar \Delta\omega$} is uniaxial and that the sample is a polycrystal (random orientation of the unique axis). In the same geometrical conditions described above the Hamiltonian for a generic orientation of the unique axis, making an angle {$\theta$} with the {$\hat z$} axis, still does not commute with {$I_z$}. Using the same description of the isotropic case, with the separation into a time independent diagonal contribution ({$n=m$}) from all time dependent non diagonal terms ({$n\ne m$}). Now the latter will oscillate at the rather high frequencies {$\omega_{nm}(\theta)=\omega_n(\theta)-\omega_m(\theta)$}, which may be observable either in a single crystal experiment (or in the presence of fast reorientations), but they are easily broadened by the polycrystalline average beyond detection capabilities if {$\Delta\omega$} is large. We can therefore assume that in a polycrystalline experiment this second contribution is missing and the polarization at very early times drops abruptly from 1 to the value corresponding to the time independent contribution. The experimental {$t=0$} asymmetry will reflect again the reduced polarization {$ (2) \qquad\qquad P_R(\theta,H)=\sum_m (\langle m|I_z| m\rangle)^2 $} where the dependence on {$\theta,H$} is implicit in the eigenvectors {$|m(\theta,H)\rangle$} of the anisotropic radical Hamiltonian. This leads to a simple matlab simulation, like the one above, where only the definition of the Hamiltonian must be rewritten. The code to implement the isotropic case with a single proton is given below, including generation of the nine spin matrices. All these programs plot just the longitudinal quenching curve, but it is easy to extend them to plotting transverse frequency amplitudes.
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