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# RelaxationFunctions

We shall treat here only aspects that are specific of µSR, referring the reader to the NMR section for the basic treatment of spin relaxation.

In brief one distinguishes:

• transverse relaxation due to dephasing of precessions, a $T_2$, or spin-spin process in the NMR jargoon, where no energy dissipation is required,
• longitudinal relaxation due to recovery of thermodynamic equilibrium, a $T_1$, or spin-lattice process in the NMR jargoon, which requires contact with a thermal reservoire (e.g. the lattice) to dissipate the excess spin energy stored in the out-of-equilibrium initial muon polarization.

The experimental condition that allows to measure separately the two terms is the geometry of the experiment: whether the initial muon spin polarization, $\mathbf{P}(0)\parallel \hat z$ is longitudinal or transverse to the static component of the local field $\mathbf{B}_0$. In order to observe a pure longitudinal relaxation we need to have $\mathbf{B}_0\parallel\mathbf{P}_0$ and the maximum amplitude will be recorded in detectors along $\hat z$. In order to observe a pure transverse relaxation we need to have $\mathbf{B}_0\perp\mathbf{P}_0$, say along $\hat x$ and the maximum amplitude will be recorded in the $yz$ plane. Examples of these geometries, including intermediate cases, have been described under magnetic materials.

The models for these relaxation functions are always derived from more general NMR descriptions (nuclei may have spin larger than $\frac 1 2$ and experience more complicated interactions; the subcase of spin $\frac 1 2$ applies to muons).

It is a general feature that longitudinal relaxation can only be due to time dependent interactions, such as those introduced by time dependent external fields (i.e. radio frequency), thermally populated excitations (phonons, magnons, etc.). For transverse relaxation one generally distinguishes the $T_1$-like component, due to time dependent (secular) interactions, from the static component, due to an inhomogeneous distribution of time independent interactions. Since the simple µSR experiment (without spin-echoes) cannot separate the two, transverse relaxation is often cominated by the larger inhomogeneous static contribution.

Muons may easily diffuse already at moderate temperatures. For instance they diffuse in pure metals, down to very low temperatures, and in oxides, already below room temperature. This, which is a condition much more rarely encountered with nuclei, gives rise to a specific time dependence of the local fields.

One very specific ability of muons is to measure spin relaxation in the absence of an applied magnetic field. Zero field relaxation may be measured also with Nuclear Quadrupole Resonance, NQR?, given nucleus with spin larger than $\frac 1 2$ and with a strong enough quadrupolar interaction. Also NMR may allow this measurement, in special conditions and with a field cycling apparatus. In this case the nuclei are polarized in a large magnetic field, which is then rapidly reduced to measure spin dynamics in low or zero field. How rapidly a large field may be varied imposes restrictions both on the sample nature and on the accessible realxation rates.

With muons the experiment is particularly straightforward, and has virtually no restrictions, since the polarazation is provided by parity violation.

 Ryogo Kubo (1920 - 1995)

A peculiar µSR experimental condition, therefore, is when only relatively weak, random magnetic fields are present, like, for instance, in the case of nuclear dipolar fields. The relaxation functions appropriate for this case were obtained by Kubo and coworkers.