Abstract
The third chapter introduces the principles of the \(\mu \)SR technique. First, its main features that distinguish it from other so-called solid-state nuclear techniques (NMR, neutron scattering, EPR) are described in detail. We explain how information about different physical properties is obtained in a \(\mu \)SR experiment by measuring the time evolution of the spin polarization of an ensemble of muons implanted in the in the material under study. We show how this is done by detecting the anisotropically emitted decay positrons. The main components of a \(\mu \)SR spectrometer are described. Data acquisition and the different experimental setups used and their purpose (so-called zero magnetic field, transverse field and longitudinal field) are discussed. The main differences between experiments with continuous and pulsed muon beam experiments are presented.
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Notes
- 1.
This acronym was coined in one of the first reviews about the use of muons in physics and chemistry (Brewer et al. 1975).
- 2.
Note that NQR (Nuclear Quadrupole Resonance) is also a zero field technique, but for magnetic studies less direct than zero field \(\mu \)SR.
- 3.
For an accurate determination, the muon site and magnetic structure must be known.
- 4.
In decay beams, see Sect. 1.8.3 the polarization is also very high \(\sim \) 80%.
- 5.
The density of muons in the sample, even with the most intense pulse, is negligibly small, so that muon-muon interactions can be completely ignored.
- 6.
Sometimes called \(\Delta t\)-\(\mu \)SR.
- 7.
Measurements are also performed in the so-called integral mode, where simply the number of the decay positrons in different detectors is recorded. This mode is not rate limited. It is often used in the integral avoided level crossing (ALC) or level crossing resonance (LCR) techniques (Roduner 1988), where it has been applied to determine muon and nuclei hyperfine coupling constants in molecules such as muoniated radicals or in muonium defect centers, thus allowing the characterization of the system and the assessment of its structure (Heming et al. 1986; Kiefl 1986), see also Chap. 7.
- 8.
The energy-averaged efficiency is \(\epsilon =\displaystyle \int _0^1 E(\varepsilon )f(\varepsilon )\;d\varepsilon \), where \(f(\varepsilon )\) is the efficiency of the detector as a function of the scaled energy \(\varepsilon \).
- 9.
The effect of a finite solid angle on the asymmetry can be written as
$$\displaystyle \begin{aligned} A_{0,\Delta\Omega}\;\cos\theta=\frac{\bar{A}\int\limits_{\theta-\Delta\theta/2}^{\theta+\Delta\theta/2}\cos\theta'\;d\theta'}{\Delta\theta} = \bar{A}\cos\theta\;\frac{\sin{\frac{\Delta\theta}{2}}}{\frac{\Delta\theta}{2}}\;\;, \end{aligned} $$(3.4)where \(\theta \) is the observation angle between \(\mathbf {P}\) and \(\hat {\mathbf {n}}\) and \(\Delta \theta \) is the polar angle subtended by the detector.
- 10.
Note that a possible reduction from the ideal value \(|\mathbf {P}(0)|=1\) is generally integrated in the asymmetry \(A_0\).
- 11.
In the literature, depending on the context, the \(\mu \)SR signal (sometimes normalized to \(A_0\)) is also called asymmetry, polarization, depolarization, or relaxation signal.
- 12.
Also named continuous-wave or cw-beam.
- 13.
High-energy decay beams retain a signature of the proton pulse because the transported muons originate mainly from a pion cloud volume with a more or less well-defined decay time.
- 14.
Note that the Larmor precession frequency of the muon spin in 10 T is 1.35 GHz.
- 15.
- 16.
The high sensitivity of \(\mu \)SR requires an active compensation of the Earth’s and surrounding magnetic fields to perform ZF experiments under ideal conditions. The compensation is usually better than \(\sim 2\; \mu \)T.
- 17.
- 18.
As remarked earlier, we define backward and forward directions with respect to the incoming muon momentum. Sometimes backward and forward are defined with respect to the initial spin polarization. With a spin-unrotated surface muon beam, as in Fig. 3.9, this means swap** the labels of the positron detectors.
- 19.
We will see later that the magnetic field of the nuclear moments produces a slight change in the polarization.
- 20.
Note that the effect of the exponential decay due to the finite muon lifetime enters in the statistical error of the histograms and of the asymmetry. Due to the Poisson statistics, the relative error of a histogram bin has an essentially exponential growth with time t, \(\sim \exp (t/(2\tau _\mu ))\).
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Amato, A., Morenzoni, E. (2024). The \( \mu \)SR Technique. In: Introduction to Muon Spin Spectroscopy. Lecture Notes in Physics, vol 961. Springer, Cham. https://doi.org/10.1007/978-3-031-44959-8_3
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